My 5550 Watt generator hunts underload and finallly cut off, but before it stops, it slows down and then pick speed a few times and then finally shuts down?

Hello, it seem that you are having a Governor problem. On the throttle side of the carburetor (second one back from the front of carb) this the throttle. There a lever/spring and rod the go back to the side o the engine block. Their is a angler clutch in the crank case. As load is applied to the generator this will slow the engine down. This will causes the governor clutch to spring slower which in turn will pull the Governor rod back---by pull back the rod this will open the throttle for more rpm.. The same with no load, the governor rod will slow down and will semi close the throttle place letting the engine/generator idle.
Now, if this governor system is working. Then look at carburetor. Their is a nut or screw in type solenoid that holds the fuel bowl on. Remove this solenoid/bolt and clean out the settlement bowl.
If it has a fuel filter then replace. Another thing here, if the is old fuel in the tank drain it out and clean out the fuel tank. Modern gasoline is good for about 4 months of storage, and after a while it gets really gunky. It is just cheap fuel with high prices. I pump Ethanol premium fuel for 18 cents/gal.---regular was 12/13 cents/gal. quart of brand name oil was 22 cents and recondition oil was a 5c/qt.

First thing, how old is the fuel?, does the settlement boil have gunky in it?, does this generator have a fuel filter? and finally does the governor working properly. I wish you luck buddy. GB...stewbison