If you have a multimeter, or a simple test light, you can easily perform a couple of checks to help narrow it down. Put the key in the ignition and turn to the On position. Turn on the truck's lights.
1) Open the flap over the connector in the trailer hitch. Using a test light or multimeter, bridge the connections in the 7 o'clock position (white in the Vehicle Side part of the pic) and the 1 o'clock position (black in the pic). This will verify that you have voltage - I expect it to work since you say some of the trailer lighting works.
2) Bridge the 7 o'clock terminal with the 11 o'clock terminal (green) to see if the running lights terminal has voltage. I suspect this is failing. If you have voltage here, the problem is in the trailer's wiring or connector.
3) Unplug the wiring harness from the back of the trailer hitch connector and repeat the tests above. Terminals should be in the same positions. If the running lights work here but not on the other side, then the problem is with that socket in the hitch, which is not uncommon. If there's no voltage to the running lights terminal here, then the wiring or a fuse/relay has a problem.
Good luck and report back what you find out.