Taillights and running lights do not turn off

Okay, to see where this issue is coming from first verify it's still a problem, now pull fuse 7. If the lights, go out it could be the headlight switch has failed internally. If you put the fuse back in and the lights, come back on, you will want to pull the headlamp switch out.
For that you disconnect the battery, turn the headlights on, then pull the knob out and use a thin tool to push the release in the slot on the knob to get the knob off. Now rotate the knob 180 degrees and push it back into place. Next rotate the switch counterclockwise to the Off position and then turn it fully clockwise to release the tabs so you can pull the switch out of the dash. Once you have the switch out. Disconnect the electrical connections and reconnect the battery. Do the lights stay off? If yes, the switch is faulty, if the lights still come on then the light circuit is shorted somewhere else.

Okay, the fuse in the holder shows power on both sides but when removed you only show power on one connection? That would be normal as the fuse bridges power across it, if you pull the trailer running light relay and the lights go out then that relay is stuck closed and sending power full time however that relay isn't supposed to power the trucks lighting, only the trailer lights. So, there would need to be a short in that system as well if pulling the relay turns them off. That or someone wired in lighting that bridges the circuits.