Sounds like a classic ground fault.. Possible locations are the sockets in the lights themselves the wires to the plug, the socket or #1 with a screwdriver is the connection to the motor home chassis from said socket.
When I got my towed and had the LP-1 Lube pump installed the kit included a tow light system, the Add-a-lamp type where they put a 2nd bulb in the tail light housing,, I actually like that system when it fits as it does on my car.
Electrially... IT is the same as what you have.
Well, suddenly I had issues when I had the headlights on or some other times, Appeared to be a feedback between the tail light and the turn signal/brake light
I up-graded the groun on the SOCKET on the motor home.. And that fixed it.
I also had a problem like that with no towed some time ago.. That was a FRONT turn signal.. Never saw that one coming.