I spent forty years on a service truck doing electrical trouble shooting. The chances that all the running light bulbs are bad is slim and none. The chances that all the sockets are corroded / bad is also very slim. I would suspect that the problem is at a connection point ,most likely the 7 pin connection . Check the tow vehicles 7 pin connector for power to the running lights and eliminate the the tow vehicle. Then check the trailer to see if there is continuity through the running light circuit to eliminate the trailer wiring. If both tow vehicle and trailer test good that would indicate you are not making contact at the 7 pin connector. I have had a similar problem twice ,once a wire was corroded/ loose in the 7 pin connector , the other was a bad pin in the 7 pin connector. Wires seldom break in the middle of a run unless subject to excess vibration or physical damage . Most of the time the problem is at a connection point.
The 7 pin connectors often loose contact tension after repeated use leading to a high resistance connection point ,similar to a worn out receptacle in a house.
Since some of the lights work it would appear that the ground connection is okay .
The wiring system in a trailer is not that complicated and with a good meter and a little luck you should be able to find the problem.