So, have you taken the socket apart on the towing vehicle yet???
Plugging in and jiggling the socket can make wires fall out and touch the side of the socket.
The docking lights are a separate problem.
Using an ohmmeter can be misleading for a novice and not much of a confidence builder.
Buy a 12 volt test light and you will have a go/no go indicator.
To chase the short, grab any piece of insulated wire and run it to the positive lead of your battery. Connect the free end to the alligator clip on the test light. Tape it up so it is insulated. Then the tip of the light can be touched to any (ground/short) and it will light.
Start at the socket and touch the ground pin, (should be full bright), touch the other circuits and see what you get.
When you touch it to the (shorted backup pin), it should light. Now you can chase the wire back until the short is removed. I suspect it is in the socket or very close by.