strange idling problem

[johnnyscrawny]

2007 Suzuki GZ250 just got it. It was well cared for, low mileage, no crazy damage at all. I know it ran well as of two weeks ago when I checked it out. But between then and when I picked it up a couple days ago the fuel supply valve was left in 'prime' meaning the carb and possibly the engine was flooded. Nonetheless it started up just fine but then died on its own after idling for less than 2 minutes. So I drained the tank, the carb, changed the engine oil, replaced the spark plug, and put fresh treated fuel in the tank. But then it wouldn't start at all (seemed clearly like a battery thing). So I jumped it with a booster pac and it started up just fine again but still slowly dies after idling for just a couple minutes. I've tried this multiple times now adjusting the idle screw and everything and it sounds great for a couple minutes (especially on full choke) before it noticeably slows and dies on its own. What could this be? Clogged carb jet? Fuel valve? I'm at a loss.