Our similar but I believe older ONAN BFA ("4KW RV GenSet") was hunting. After I had the carb professionally rebuilt. The manual's flowchart included "replace governor spring" after a number of other steps. The Cummins/ONAN shop was on my way to work so I stopped and got a spring. Installed and no more hunting. At the shop, they explained how the spring sits stretched year after year and can get just a pinch out of calibration.
I didn't have to adjust anything. If your genny is plain old fashioned like ours was, I don't think you'll have to either. If you get the carb mixture settings right, you might want to adjust Idle RPM. Manual says to go by 1800 RPM or 60 hZ. On the BFA, setting idle based on AC voltage accomplished the same thing. I think I set 120 VAC at no load idle. When I say "idle" that means the Throttle Plate is in Idle position and the Idle circuit of the carb is delivering the fuel. When you add load, the governor (and that Spring!) hold the RPM, but the carb throttles up and the power circuit is delivering fuel.