Hello, what is the full model number of your genset?
Depending on the model number, your set may not have points, but an electronic ignition system.
OK, first. your unit is a Spec. J. It does not have points, but an elctronic ignition system. If the engine is running, there is nothing wrong with it. It either works or it doesn't.
When you say you bring the RPM's up, are you doing it or is the genset doing it? This unit does not have variable speed. It runs at one speed (3600RPM) and stays there. It will fluctuate a little under heavy load, but not much. Do not try to make it run faster, as you can burn out some very expensive parts. What exactly did you do to the carb? Be very precise, because it sounds like it is set too rich. It also sounds like the governor is way out of adjustment. How many volts is the generator producing? At this point, a frequency test is the best test. Does your meter measure frequency? If not, do you know anyone who has a frequency meter that you could borrow? If I knew the voltage and frequency, it would really help.
Did you also replace the spark plugs when you tuned it up? If not, get a set of Champion plugs for it. I have seen many Onan's not run on other brands for some reason (I don't know why).
Let me know what you find.