OP Update:
I think I have mostly figured this problem out.
It looks as if all of the back light grounds are routed to the one point. From the actual light bulbs, the ground wires pass through a connector, combine at a splice in the wire harnass, and one large ground wire emerges from the harnass and is bolted to the frame.
All three of these were getting old and were somewhat compromised but still worked before the towing incident. It looks as if the batteries of the TV and RV are directly connected upon plug-in. The bad batteries in the RV stressed this ground connection.
I was getting all kinds of 12 volts at the back lights.
I stuck a pin in one of the ground wires of a light bulb and touched the pin to the frame and the light came on.
I cleaned up the ground terminal and bolt, but the problem persisted.
Then I got the ohm meter out and found the splice. The splice was not working.
So I peeled off the tape and saw shrink tubing with weather goop around the splice. I cut all of this off.
This exposed one of these "modern" solutions. SOmehow they had four wires and they interleaved all of the strands together. Then they did a giant crimp action smashing it all into a little brick of wires. No solder or anything.
I have a 1949 Sylvania TV. Still works. All soldered.
So I heated that brick up and doused it with solder. Fixed.
Then the lights came on. But then the next day they went off again.
So I jiggled the connector and saw a light flicker.
Upon inspection, the ground pin of this connector was cooked. I cut it and it now temporarily has a wire nut on it.
I may just clip out this connector and solder all of the wires together.
This may also be the problem with the blinkers and back up light functions.
I saw evidence of heat damage to other pins as well.
