Draining bowl can help and it "should" fill back up but the float can stick, the float valve can stick, the fuel pump can be weak, you can have a leak or pinhole in the fuel hose from the vehicle tank.
Draining bowl does not drain all fuel, it will typically leave a small amount in the bottom of the bowl unless you remove the bowl and dump it..
That leftover remaining fuel as it evaporates off tends to attract moisture from the air which often will corrode the fuel bowl.. The corrosion sits on the bottom of the fuel bowl until you fill the bowl and then it gets picked up by the jet and clogs the jet and or passages.
Hunting means you typically have a starvation fuel problem (IE lean condition), if you have a manual choke you can sometimes "correct" lean condition by closing the choke until smooth.. But that is not a real fix, just a bandaid plastered on top of the problem.
You need to find out if you are getting enough fuel from the tank (weak pump or fuel line issues).
If you have good fuel flow then carb cleaning most like is needed.. If I remember correctly, Onan doesn't recommend cleaning the carb, only replacement and they are not cheap and a few models impossible to source.. I would try removing the carb from the engine, breaking it down and remove and clean the jet and passages first if you have plenty of fuel flow from your tank..
On edit, I should note that on some gens the fuel pump may be electric and some are vacuum "pulse" pumps which use the intake vacuum pulses via a rubber diaphragm.. The diaphrams often get softened from the gas, sometimes they get brittle and brake or they may get a pinhole causing the fuel pump to not work correctly..