Take apart the 7-pin Pollak-type trailer connector on the rear bumper where the wiring harness enters the back side of the connector and clean it out thoroughly. You'll probably find it packed with mud, road grit, water, etc. When you reassemble it, pack it with dielectric grease to keep out any contaminants.
I experienced a situation such as you describe with my previous 2002 Ram 3500 that used the Jordan Optima 2020 brake controller. I would go out in the garage and see the "trailer connected" light glowing dimly on the controller. This was caused by a high resistance short between the +12VDC wire and the trailer brake wire in the 7-pin plug due to water and road grit contamination. As far as discharging the battery, there could likely also be high resistance continuity between the +12VDC and -12VDC (ground) terminals if the discharge rate is very rapid. After cleaning the connector out and packing it with dielectric grease, I never had another problem.
Rusty