Sounds like , either fuel or electrical issues. Have you checked all fuel line connections and filter along with carb cleaning (jets) If so then you may have a igniton module about to go south . Usually a bad coil will rerquire a few mintes running before it may heat up and cause your form of symtoms. Also you may wish to check your on / off switch they sometimes will cause sporatic on off type of issues that you have described .

I was going to try to clean the carbs next. The on-off switch is interesting. Is there an easy way to test that? Does it just ground the ignition module when Off?

Thanks

The quickest way to test the on/off switch might be to just try wiggling the switch a bit while it's running? Tap it, etc, and see if the problem changes. I would imagine that it grounds out the ignition when it's engaged.

Alternately, along the same lines, perhaps you have a wire to/from the switch which is rubbing against something metal, wore through the insulation, and is sporadically grounding out the ignition by itself?

Rock: Yes as red has indicated you could try the simple wiggle test when the saw begins to demonstrate the symtoms again while running . Another way is to attach you multi meter to the switch and check for continuity (ohms) A open circuit in both on and off would indicate a defective switch or as been advised you spark plug wire could be grounding out also . Do a inspection if all tests out then give the carb a good cleaning and see what happens .

I checked the switch with a multi-meter and it looks good. Unfortunately I also disassembled some of the saw, including the carb, and I messed something up. It's not pulling any fuel in now. I think I broke one of the carb gaskets. Darnit! I'll order a small gasket set and see if I can get it back into shape. I don't want to sink too much money into it if I can't get it going. We'll see.
Thanks

Rock: While your at it get the diaphram kit , it is not much of a $$ adder. A weak diaphram can cause poor performance also . Older saws with a little dirt or varnish along with a weak diaophram will cause a lot of headaches. Ensure to check all your fuel lines for tightness a small air leak can cause serious damage long term. Also as previously suggested , ensure your fuel filter has no restriction you should be able to blow through it . If all is good then just a jet orifice cleaning with a concentrated fuel cleaner should get you going :)