I have diagrams for a 96 and a 99. The problem, is they changed these trailer tow circuits a lot, almost every year, so I am afraid to use one of those diagrams to troubleshoot your truck.

But through the years, they always seemed to use a relay for the tow marker lights. I did look on Autozone's sight, and their diagrams are not very complete for your year, but they do show a marker light relay for towing.

Are you positive your wiring in the rear is hooked to some official trailer tow wiring? You should have at least one relay somewhere for it I would think.

Maybe someone has some info on a 97 and can write in. My goal for you to take the relays out, was to kill all power to the rear trailer tow wiring, and see if you still had voltage on the wires with no lights on. If you did still have voltage readings like you do now, you have a serious wiring problem with the tow wires, and this is not a normal failure for these circuits.