Thanks for the help! I discovered there is an outlet for the generator where the shoreline cord is. I plugged the shoreline into that outlet, started the generator, and everything works. I ran the A/C and the voltage stayed at 119. I ran the microwave for a few seconds with water in it, the generator barked and kept going. I feel like an idiot for not knowing the transfer is human actuated, but this is how you learn.
As for the Onan 4.0, it needed everything- circuit board, fuel pump, and I rebuilt the carb. I also made new spark plug wires. It runs great and has only 57 hours for being 30 years old. I think the circuit board failed in the 90s, IMO.