Hi Engiman34,
Welcome to the Heartland Owners Forum.
Running and tail lights are not powered by the trailer battery, so if they're on, it suggests an alternate power source.
If the trailer is plugged into shore power when this happens, I'm wondering if perhaps you have a "hot skin" condition where the trailer exterior is carrying power. This would be usually be caused by 2 failures: a 120V device that is leaking power, along with a lack of ground connection where the trailer is plugged in. Sometimes using a damaged extension cord, or plugging into an improperly grounded power receptacle, will contribute to this condition.
An easy test: when you see that the lights are ON, turn off power to your shore power receptacle, or unplug the trailer from the receptacle. If the lights go out, the next step would be to get a $20 Non Contact Voltage Tester from a local hardware store (skip the $10 testers). Plug back in and if the lights are ON, see if the tester indicates a "hot skin" as you approach the trailer.
"Hot Skin" can be lethal, especially if the ground is wet. Be careful about going into the trailer or touching the trailer until you eliminate this possibility.
Alternative theories on your lights would be 1) the previous owner changing the wiring in some way, 2) a cross connection, probably at or near the trailer umbilical cord.