If it is a factory plug it should be fine, but if an aftermarket one, that has a couple of screws holding the fat end of the plug together then do take it apart and see if any little wire strands have escaped and visiting the neighboring wires. It is only the 12V positive wire that can make the lights come on. It is possible to pull wires that are otherwise fine out of place when taking the plug apart, so beware you could double the problem if not careful.
Otherwise, as mentioned, you likely have a short within the bowels of the camper. To run new wire from where the trailer/truck plug is wired to the camper (near a fuse box) and all the way to the light... to every single light around the camper will be a big job. Hopefully the short is either in the plug head or right there at the fuse box area.
First check the tail light bulbs and wiring, a short here is easier to fix. Look behind the fixture that holds the bulb to see if its wire is shorted.