Start by suspecting the fuel filter, since it's cheap & easy, and fits the symptoms well. If it's running good while it runs, it's probably not a gummed up carb, but Seafoam or Techron won't hurt.
That leads to three other possibilities: clogged spark arrestor screen in the exhaust (doesn't quite fit symptoms, but can be DIY'd), failing fuel pump (fits symptoms, $$, and generator has to come out), or voltage regulator (somewhat fits symptoms, $$, can be DIY'd, troubleshooting chart in Onan manual & on Flight Systems website).
Jim
'06 Tiger CX 'C Minus' on a Silverado 2500HD 4x4, 8.1 & Allison (aka 'Loafer's Glory')