Now that problem sure sounds like something I had when I hooked up my towed for the firs trip.
Here is what I found.. I had the Remco LP-1 Lube pump installed on the towed (minor issue, easily fixed with the product) and the instrucitons show it coming with a diode kit for the towed lights.
It came with an "Add-a-lamp" type system
I was getting what I call "Feed back" between the tail lights and the brake/turn lights causing just exactly the symptoms you cite, also interfering with cruise control operation.
Well, the six pin connector connected to the motor home using a device not unlike a "Scotch Lock" connector (it is an insulation piercing thing that clamps on the wire)
I stripped back a bit of insulation on the socket side of the scotch lock (Made very sure I had the ground wire) SOLDERED on a new length of wire and sealed it up good.. Ran the new wire to a good ground connection near by.
POOF went the problem.. It was a bad ground, Motorhome side.
NOTE: There are many canididates for bad ground. INCLUDING the lamp in the socket.
I had a problem much like yours when NOT towing..
Turned out to be one of the turn signal lamps not grounding properly in the socket.
THIS.. Is exactly what WD-40 was invented to fix.. and it did.