If I had to guess, and that's all I can do from here
I'm not sure what you are calling "running lights, I'm going to assume that you mean the clearance lights...the following is based on tail lights out, clearance lights are working fine.
Since you have power on the TT side of the plug, you've eliminated that as a source. Follow your 7 pin cable under the front and into the storage area...you should see a junction box (mine is black plastic) and your 7 pin cable will go into that and then branch off for the various circuits in your trailer, several of the splices will go to a standard 4 wire trailer wire cable (R turn, L turn, brake and running *you'll note there is no ground in that bundle, that's coming
). Check those connections, but, it would be odd for all four of them to be bad. If you look under your trailer, you should see that four strand, standard trailer light cable running along the street side frame, on the inside edge. Follow that back to look for breaks in the cable. If that is "solid" (not broken/torn) then the problem will be with the lights themselves. Corrosion is a possibility, but, not likely on both tail light assemblies at the same time. That leaves the ground path...the lights are not grounded by that 4 wire harness from the front, they are grounded from the light sockets to the frame on a separate ground wire, all the functions for each light are dependent on that one ground wire from each light and both are usually grounded at the same point, if that ground point is damaged (broken, loose, corroded, etc.), none of the tail light functions will work. I suspect you'll find the problem there...
Best stab I can take it from here
BTW, I can appreciate the bugs you've got...we just got back from a couple of months on the road that included a few days in New Iberia. Lovely country, but, yep, you've got some "killer" bugs...I've got a grill full of their little carcasses!