Since you mention brakes I assume you are working with a 7-round plug. You also mentioned teardrop so the trailer should be light enough to safely tow without brakes.
Quick solution for towing now and solving the real problem later:
Buy new lights and wiring combo. This should include the lights, all wiring, and a 4-flat plug. All of it should be connected already so all you need to do is ground the white wire, find a way to mount the lights, and zip-tie the loose wiring up. This should get you to where you need to go without any problems. You can easily remove everything at a later time and place to work on the actual problem.
Later:
Since all the wiring is preexisting and just the housings are rusted, you should be able to remove the original lights and housing to expose just the wires hanging out, most likely 2 wires. Find new lights that match what is hanging out of your trailer, 2 or 3 wire. Some require grounding, some do not.
