I second KTSOV with his question. Does your Equinox have the 2.0T engine that came with the hitch and tow wiring? If so, take it back to the dealer for warranty service.
If it was wired by the RV store, the wiring harness they used plugs into the light socket, and then the harness socket to the light. They did not get the socket sealed correctly, and moisture is getting to the taillight electrical system or into the trailer plug causing the issue. Was it installed on the exterior of the vehicle?
I installed my 4-way plug to be inside the vehicle and just hang out the hatch when needed.