You might try inspecting the connections in the junction box in the A-frame where the umbilical cord wires splice to the trailer's wiring. The connections are exposed to water, dirt and maybe salt from when it was originally delivered plus sometimes the factory does a really poor job of the connections. On our TT, the running lights have a ground and positive connection with a 2-wire cable in the junction box and there is a common ground for all circuits to a ground lug on the frame. Check that all the ground wires bundled and spliced together under a wire nut inside the box is good. If you don't have one, I'd get a voltmeter/multimeter to help in diagnosing.
You could always open up the splice to the marker lights and see if you can get them to come on by jumpering to a 12 volt power source.
A typical TT wiring diagram is below. They can vary between TT manufacturers so don't use this as a standard diagram. Ours does not use the center pin #7 for ex. Disconnect the TT battery before playing around with splices in the box.
If you do end up going into the junction box and find poor connections, you may want to think about redoing all of them. Some (like me) have moved the connections inside a watertight box and used better wire connectors. I had the main positive wire fall right out of a ring terminal when I touched it which could have had serious consequences.
