Running lights will not turn off

Welcome to 2CarPros.

The first thing I need you to check is the taillight relay. If the lights are off when you reconnect the battery, then it isn't a typical short to power. Something is getting power (like a relay) and then not releasing to turn power off. Here is a link that explains how to test a relay. Additionally, I have attached a few other links to help with the basic testing equipment needed. If you find there is another relay in the box with the same part number, switch them to see if that changes what is happening.

https://www.2carpros.com/articles/how-to-check-an-electrical-relay-and-wiring-control-circuit

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

https://www.2carpros.com/articles/how-to-use-a-voltmeter

https://www.2carpros.com/articles/how-to-check-wiring

If you look at the attached picture, it shows the relay and location. Do this and let me know the results.

Joe