The #2 suspect is probably the taillight harness. I doubt that's your problem, since you're seeing behavior change when another circuit gets power. More common failures on the taillight harness are the ground wire coming loose from the socket or the contacts getting hammered flat with age...
The #1 suspect for weird taillight behavior is the trailer wiring harness. It's behind the right, rear quarter panel trim. Just fish it out -- it's a small box -- and remove it from the circuit.
Edit: to be a bit more clear, pull the large plastic trim piece off, then stick your hand in and...