Well, #3 is out. I just removed the welch plug, cleaned out the jets behind it (they weren't too dirty) and put everything back together. Unfortunately the surging remains.

I thought of another potential cause though - I noticed that the spark plug wire coming off of the ignition module is looking pretty worn and cracked. Could this cause the surging?

***Doubtful the spark plug cable has any bearing on the surging, still you might run the engine after dark to see if any errant arcing is taking place.
***Check for leakage between carb and head by spraying carburetor cleaner (very lightly ) directly on the gasket seam between carb and head while the engine is IDLING.
IDLE SPEED is the key because engine vacuum between the carb and the cylinder is highest at idle because the engine is sucking against a closed throttle plate.
Keep the air filter in place for this so the evaporating fumes from the carb cleaner are less likely to be drawn into the engine through the carb, which would skew the test.
What you are looking for here is a change of engine RPM that you can relate to a spray of carb cleaner that will be drawn into the intake through any vacuum leaks at the gasket seam. At idle, the least amount of cleaner that is inhaled through a leak will alter the RPM.
I feel more like the surging might be caused by wear in the carb linkage connecting to the governor. Excess wear makes a little movement become exaggerated when the governor tries to compensate and that can cause the kind of wild over-reaction you are seeing.
If you can detect 1/16" or so of free play in any of the link rods connecting the carb to the governor, this may be the cause of surging. If you have several individual links in the set-up, the additive wear of each link point will contribute to the total wear sum. I have seen people address the wear/free play by connecting a weak pull type coil spring along side a link (or by running the link through the inside of the spring coil) and the hooking the ends of the spring into the same hole as the end of the rod or link. The spring needs to be strong enough to keep the slack out of the linkage, and it does not take that much force if the throttle shaft is free to move in its bore.
Which brings up another thing that can cause hunting and surging: If the throttle shaft binds any at all where it goes through the body of the carb, it will cause surging.
Check this by removing the connecting linkage from the throttle shaft bellcrank arm (lever on throttle shaft) and "flip" the bellcrank with your finger tip. If the throttle shaft does not literally bounce back and forth off of the stops, it is too tight and can cause surging.
I have seen cases where the throttle shaft had to be removed from the carb body to clean up the bore and the shaft (especially likely if carb has been stored in humid conditions a long time).

Thank you very much for the tips, I will try those out.

Regarding the fix for the governor using a coil spring, do you happen to know where I could find pics of this? I'm having trouble picturing what that would look like.

Thanks again

I went out and looked at the governor linkage and there is a little play in there, but I'm not sure if it's 1/16". Either way your spring trick sounds cheap and easy so I think I'll try it out.

One other thing I forgot to mention is that the motor backfires during the surging and once when shut off. I am able to reduce this by adjusting the carb but I can't completely eliminate it. I'm not sure if that changes things or not but thought I would at least mention it.

Here is a very generic drawing to illustrate how an "anti-slack spring can be configured in conjunction with a solid link. By drawing the throttle shaft lever and the governor arm toward each other, the apparent slop or free play (lost motion) is eliminated.

{{gwi:338511}}

Though economic woes abound, today's best bargain remains...........a picture. Pictures are still worth 1,000 words each.
If you can post a close-up shot of the carb showing the adjustment screws, we might be able to get you on track to a proper adjustment of the carb.
It just occured to me that this carb might have an "adjustable main jet" and if so, it must be adjusted correctly by procedure. Say cheese!

Sure, while I was at it I also took a pic of the governor linkage so that you could see that too. Also, I got a nice surprise when I took the air cleaner off as gas spilled out. I suspect that the gas was shooting out the back of the carb as it was running because I believe that I have the float set up correct and it was sitting for more than a day and didn't leak a drop of gas on the ground.

I believe that I calibrated the carb correctly. Here is what I did:
1. Start with the idle and main mixture screws at 1.5 turns out.
2. Start the engine and set it at full throttle until it warms up.
3. Adjust the main mixture screw until the engine is running the most smoothly.
4. Turn the engine down to idle.
5. Adjust the idle mixture screw until the engine is running the most smoothly.

One issue I had with this is that when I adjusted the main mixture screw so that the engine ran optimally it was like 3 or 4 turns out and was still surging and backfiring a bit. Then when I engaged the blades black smoke would shoot out of the exhaust. So I just calibrated the main mixture screw to where it ran the best with the blades engaged and didn't blow black smoke which was around 2 turns out. Again, the engine doesn't surge at all when the blades are engaged.

Pics below.

Thanks again for your help, I'm learning a lot that I didn't know

Top View:
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Side View:
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Linkage:
{{gwi:338514}}

The procedure you used for setting the main metering screw (adjustable main jet= screw needle on bottom of carb bowl) was acceptable. Were it not for the surging, the following procedure would work.
(1)Lightly bottom the main metering screw to its seat (turn CW).
(2)Next, turn the screw CCW 1.5 turns.
(3)Start engine and run til it reaches operating temperature.
(4)Holding throttle in wide open position (max no load RPM), continue turning the Main metering screw CCW until the engine just begins to lose RPM, or you see black smoke in the exhaust (or both).
(5)Now, begin turning the screw CW and carefully count the number of turns you make as you go. When the engine RPM once again begins to drop, stop turning (you can stop the engine too).
(6)Now, turn the screw CCW exactly half the number of turns you counted. This is the "median adjustment point" for fuel volume.
The fuel metering must now be "fine tuned" to the working conditions (loaded operation) by actually putting the engine to work and then reading the color of the spark plug afterward.
This is done by running the engine under normal conditions (mowing, in this case) for at least 10 minutes and then cutting stopping the engine and removing the plug for a color reading. If the plug color is somewhat white or light gray, turn the main metering screw 1/4 to 1/2 turn CCW (more fuel) and run the engine for 10 more minutes under working conditions and then read the plug again.
The optimum color to attain is in the light brown to tan range.

Finding gas "puddled" in the air filter housing might mean the intake valve is leaking some compression past it.
I don't believe you can be certain the float level is correct based simply on looking at the ground under the carb after it has sat inert for a while. Unless the carb has an atmospheric vent to the outside, any overflow from the bowl will drain into the cylinder (but you have not stated seeing any of the typical signs of that either).

Looking at your pics, there are 2 link holes in the throttle shaft bellcrank where the link is now connected (the link is in one hole). Are these 2 holes the same diameter? or is one bigger? Any chance you might have put the link back into the wrong hole?

In the other pic, I see the throttle shaft has a return spring around the shaft. Check that this spring is actually functioning and not broken (with the remnants of the spring still in place). See both edited pics below.

Have you checked the engine valves for proper adjustment?

{{gwi:338515}}
{{gwi:338516}}

1. Regarding the throttle return spring, it does appear to be intact, but the little 90-degree 'kink' on the bottom of the spring that I believe is supposed to hold it in place is able to move around. I'm not sure if I described that correctly. Is there a hole or something in the carb body where I'm supposed to insert this? I'll also remove the linkage and check the function of it when I get home.

2. I know for sure that I put the governor linkage in the same hole it was in when I got it. I can also see that the hole it is currently in is a bit more worn than the other hole. Both holes appear to be the same diameter. I'll try connecting the governor linkage in the other hole and see if it makes a difference.

Also, looking here the throttle linkage on page 84 looks pretty similar to mine and the governor is using the same hole. Though I suppose since it's a different motor maybe it's irrelevant.

Ok, I tried your last couple of suggestions and have some observations:
1. The throttle return spring was actually too loose, and wouldn't even return the throttle all the way back to closed. I think this is due to the kink on the bottom of the spring not being seated in anything. I moved it so that it seated enough to close the throttle, but I'm not sure if it's seating in the right place. This didn't seem to affect the surging.
2. Despite what I thought earlier the other hole in the plate is actually too small so the governor linkage is definitely in the right hole.
3. I took the air cleaner off and a little more gas seeped out (and the engine hasn't run since last time I took it off) so, despite my assumptions, the float needle clearly isn't seating properly and there is a small leak. Since the intake manifold actually goes up against gravity, the only place for the gas to leak is into the air cleaner. I'll have to pull the carb bowl off and take a shot at getting the needle to seat properly. Any suggestions for this would be appeciated.
4. The choke linkage was a little bent and was preventing the choke from completely closing. I fixed this but don't think it had anything to do with the other problems.
5. I ran the engine for a bit with the air cleaner off and I noticed that during each very brief 'lull' in between surges, that most of the time (probably 75% or so) a shot of gas would shoot out the back of the carb each time. It almost seems like it's missing rather than just plain old surging. Is it possible that the intake valve is loose enough that it is periodically not fully closing and causing the surging?

I know I've said it a few times, but thank you so much for your help

I know one other thing is that these are solid lifter motors so the valve lash can get way out of wack. Maybe it would be worth my while to pull the valve cover and re-adjust the lash to see if that helps?

Yes, I suggested you might check the valve adjustment in the last sentence of my post yesterday.
I'll get back later tonight. Got a ton of stuff to do before I leave for work.

Will do. No worries, I've got a wedding coming up for the next few days so after tonight I won't have time to work on it for a while. No rush.

Thanks again

Ok, tonight I went out and re-adjusted the valve lash. The valves were a little loose, but not too bad. Anyway, it didn't seem to make much of a difference.

I still haven't tried checking for a leak between the carb and the engine. I had planned on doing that tonight but I got a little woosy from the fumes so I decided to call it a night.

Thanks

Hey mownie do you think you could post some close up pictures of the rear governor set up, the plate that connect to the governor arm.

I do not have any pictures to post showing that. Those pictures above were not mine, I only copied them from brianjohn and edited some things into them and reposted the edited pictures. Sorry :^(