What does "power at the plug" mean? Are you using a meter and your checking the running lights, tail lights feed on the vehicle side at the connector?
There's a way to use a spade type fuse to jump across the trailer connector to illuminate the RV running lights. Every now and then someone will ask how to turn on the running lights while unhitched. If you could find one of those posts and follow the instructions it would help to find out where the problem lies.
I just had this problem and found a blown 15 amp fuse for the RV tail lamps on the truck.
