My 4000 had similar problems. It also would not run at full load, i.e. running the microwave and AC at the same time. If I started one or the other the gen would slowly bog down and die within 20-30 seconds.
After having a Cummins Onan tech look at it, it turned out it was running too rich. He suspected it had sat for awhile and accumulated some debris around the needle valve and the previous owner kept opening the valve to compensate. Once the debris cleared out full flow was restored and it ran way too rich. It took several adjustments over a month or two, but it is running fine now.
To adjust the needle valve, you have to remove the "altitude adjustment" nob (mine is a black plastic ring set over a brass thumb screw.) Pry the ring off carefully using two flat-head screw drivers on either side. Then turn to the right to reduce the air/fuel mixture. I think mine ended up being almost one-and-a-half turns by the time it started running right.
Always make sure to run it about an hour under at least half load once a month. I've forgotten and not done it for almost two months and been fine.
Hope this helps!
