I had the same problem with the new aluminum trailer I just built. My brakes and turn signals worked fine, but I had no running lights.
Here's a breakdown of my diagnostic, and how I fixed it. Found a blown fuse in my trucks' electrical center. Replaced it, popped again as soon as I turned on the running lights. So, I knew it was a short to ground. Brand new harness...WTH?
OK, so, next step was to use the continuity meter to check for the short. I put one probe on the ground (white) wire of the 4 pin plug, then touched the other 3, one at a time, and got a beep at each one. Huh? :?: That can't be right!
OK, let's check a known working trailer harness, so, I checked the trailer for my Triton, got the same thing. WTH? Then it dawned on me, the light bulbs have to be removed to take this reading, as they complete a circuit from hot to ground. D-oh! #-o
Allright, so, I removed all the bulbs, then started testing. Got no continuity on my turn and stop circuits, but got a continuity reading on the parking lamp wire. I left it connected and jiggled the wires around, to see if something was touching. It continued to beep. So, the next step, I started unbolting a side marker while listening to the continuous beep of the meter. As I removed the stud from the trailer frame, the beep stopped. That's the short!
When I looked at the back of the marker lamp, I realized where I screwed up. It has little holes where you press the wire in. Well, in addition to having a slot for the positive wire from the brown wire of the harness, it also has a slot for ground. Looking inside the lamp, you can see that one hole comes through and touches the stud, where the other one is not in contact with it.
Not sure why they do this, unless it's for non-metallic frames that need a seperate ground wire for each lamp. Anyhow, in my haste or inattention, I stuck the wire into the ground slot, which caused a short. Removed it and put it into the correct slot, then checked continuity, no beep. Re-installed the bulbs, and turned on the parking lamps...guess what? Fixed! =D>
It's possible that it's something like this causing your issue.