I want to thank everyone for their input on this issue. Yeah Puddle Pirates, that's what I was thinking too. I finally got the furnace out and apart. NO easy task for any of it.
The gas valve coils checked out ok. The sail switch seemed to be ok. The ignitor seemed to be ok, but not sure how to test that really, nor how to test the motor. By what I've researched, even though there's no house battery, and I don't think it ever had one, someone could have wired an exterior battery or generator the wrong way at some point before I acquired the camper; reversing the hot and ground coming in to the camper. Either that, or, yeah, a bad plug. And/or maybe a bad motor even, as someone said.
Anyway, since the furnace is at least about 19 years old -- although I don't think it was used much as the burn chamber looks hardly used -- and since I have it all torn apart, I decided to replace some things because I sure don't want to have to go through all this again. So, I'm awaiting a sail switch, a motor, a field plug assembly, a limit switch, an exhaust wall gasket, and an electrode. I'll double check the tracing of the wiring going from/to the inverter in the mean time. But once done, it should be like a new furnace. I'll update once I get it all put back together and it's running.