Stabil and other fuel treatments are good products and we recommend their use in all ethanol blended fuels. However, as the gas evaporates, it will still leave behind a wax which will clog things up.
I would start by removing the main jet that comes up through the bowl and spraying a little carb cleaner inside and running a fine wire through the holes. Then replace the jet, screwing it in untli it bottoms out and then back off 1 1/2 turns. Try the engine at this point and see if there's an improvement.
If the engine will run, then set the engine to operating speed, and adjust the jet by screwing it in until the engine starts to run rough. Then back it out until the engine smooths out and then begins to run rough again. The half-way point between running rough is the proper setting. Then adjust the jet on the top of the carb, the idle mixture, the same way except set the engine to run at a slow idle.
If this doesn't solve things, then I'd pull the carb off and give it a very good cleaning. This carb you can usually get by with spray cleaner, fine wires and compressed air, but soaking or ultrasonic cleaning are always better methods. Be sure to replace, not try to clean or just blow out, the fuel line and filter and clean out the tank. Often it's the fuel line breaking down and sending little chunks of rubber and a black goo that clogs up a carburetor, so replacement is the way to go with fuel lines and filters.
Another thing we recommend on L-head engines is cleaning the head. You can pull the head fairly easily on the engine, I don't know what generator parts or frame might be in the way. But carbon will build up and needs to be removed. Some fuel additives even add to the carbon build-up. Excess carbon can cause a problem like you have and removal is a normal maintenance item on L-head engines anyway. Just be careful not to gouge the head or piston when scraping and brushing the carbon.
These are the first things I would try and I think have a good chance of catching the problem you have. If you don't see an improvement from this, let me know and we'll have to go to less likely items. Thanks, PK.