I'm sure blinkers have been modernized. The replacement I bought at NAPA several years ago had a chip in it and no bimetallic strip.
Monitoring the voltage across one of the lights might be informative. It would be interesting to check the voltage from each side of a misbehaving bulb to ground. "It happens when I attach the TT" certainly suggests something in the TT is feeding a small current back from one lighting circuit to another. Yes, I think a grounding problem could be the culprit. I would find a good ground on the truck and use a wire to connect to a good ground on the trailer for starters. Or even use the ground wire to ground one of the misbehaving lights to see if it has any effect.
Think of it as an interesting puzzle to avoid impatience and maybe you can enjoy figuring it out. Please let us know how you solve it!
