IMHO, you have an injector nozzle the is leaking down into the cylinder
while it is stopped.. Then that injector is dumping raw fuel into that
cylinder while running.. (Hence the black carbon and oily film in the
exhaust).

Until you get the engine speed stablized the generator can't settle down to
the required frequency or voltage.


--
My opinion and experience. FWIW

Steve
s/v Good Intentions