I know it's a shot in the dark but you might get a new piece of fuel line a few feet long, and connect it to the Generator and put the other end into a jar or coffee can with gas. If it sucks that out and runs great, it will rule out an air leak in the fuel line between the genny and the main fuel tank.
If it still wont stay running then you may very well have a carb issue. I've had to disassemble and clean the carb on my brother's onan every other year or so. I'm just careful with the gaskets and clean with a spray can of Berrymans carb cleaner and blow out with compressed air.