You have a gravity feed fuel tank AND the needle valve in the carburetor is not shutting off the flow of fuel AND the fuel tank emptied into the engine crankcase AND when you started the engine, the gasoline diluted oil foamed AND got sucked out of the crankcase to be burned in the engine.
I wonder if the carburetor on this engine was transferred from the other engine that "ran out of oil"? If so, the score is now Carburetor 2, Engines 0.
Will the engine turn or is it locked up?
You might get lucky because sometimes the old "gas in the crankcase" event leads to such a fuel rich mixture going into the intake as the engine heats up and the gas begins to boil off, that it smothers the engine causing it to quit running. You need to hope that's what happened, or that the fuel tank just ran dry before the engine could lock up.
Anyhow, determine if the engine will still turn.
If it will, do the following in this order.
(1) Change the engine oil (and filter if it has one).
(2) Take the carburetor off and remove the bowl, float, and needle valve for a good cleaning and inspection. Clean out the entire carb using spray carb cleaner and poke a small stiff wire into all holes and passageways, followed by more carb cleaner. Compressed air is good to blow out all the passages if you have it.
(3) Purchase an inline fuel shutoff valve, mount it between the fuel tank and the fuel filter.
(4) Replace the fuel filter and fuel hoses.
(5) Remember to always turn off the fuel shutoff valve whenever you cut the engine off. Get into the habit of doing that.
If the engine is locked up..........perform the above steps 2 through 5 to your new engine before you run it

To add to the above I would check the oil for: overfilled if so smell for oil for gas? (tail tail sign of gas in oil or you overfilled it?)(I alway like to know what happened and if I know that I can determine what I need to do)

Then I would Remove the spark plugs and see if it will crank over using the starter, check for find mist shooting out the spark plug holes (another tail tail sign of gas in cylinder or cylinders and hydraulic lock). I spin the engine over till blow out all the residue, then I spray carb cleaner down the spark plug hole to wash out any oil film so when and if you get it going again if will fire right off providing the above don't take too long.

So I drained the oil, or should I say gas....replaced the oil filter, added new oil, removed the spark plug and turned it over a couple of times. There is already a shut off valve after the gas tank so I did not need to add that. I just need to stat using it. I opened up the carb and it doesn't seem like anything is sticking or that it is gummed up. But i'm not really sure how to tell. The air filter became saturated with gas when this all started happening, so I am sure I will have to replace that. I don't have the fuel filter yet so I wasn't able to replace that.

Anyway, after I did all this I added a little bit of gas and tried to turn it over. It started right up with the same big puffs of smoke. I turned it off and put a towel under the muffler which is still dipping a fuel/oil mix out of it. I tried starting it up again and it started righ up again in the same manner. I tried turning it off and on a few more times and had the same result each time. The last time it started with out smoke but started to smoke about 15 seconds after being started.

Do you think that I will just need to let it run for a while to burn off any oil that might still be in there, or do I need to continue the process that mownie described and either clean or rplace all the hoses and what not.

Thanks all!

OK, good, you are making progress.
When I say turn off the fuel valve when it's not running, I mean that literally. In fact, I recommend you turn off the fuel valve and let the engine run for a few seconds after the fuel is off before stopping the engine. Doing this will lower the fuel level in the bowl a bit.
Lots of times you will not be able to point a finger at something and say, "That's why it was leaking fuel into the engine." It could have been a tiny piece of crud that got stuck under the needle valve keeping it from shutting the fuel off completely. Just learn to get in the habit of turning off the fuel (even if it means making up some kind of sign) and as long as the engine runs well and has power, you can postpone any further carb cleaning.
I advise you to begin adding some Marvel Mystery Oil and also some Seafoam fuel treatment to your fuel supply (from now on).
Keep your eye on the engine oil level to make sure that the present fuel valve does in fact cut off the fuel.
As far as running the engine, go ahead and mow with it. Yes, it will spit out some drops of oil and flakes of carbon but that will end after 15 minutes or so of working the engine under a normal load. You will want to check the oil before each use, or when you refill the fuel tank. This will insure that the engine will not run out of oil if it has suffered some minor scoring of the cylinder walls during your adventure. If the engine continues to blow smoke and consumes a lot of oil you may want to consider some repairs to the engine, but let's see how it does first. Good luck.