Thanks for the more information. Many of the newer trucks have independent fuses for trailer towing. I would definitely look at the fuses before removing the plug.
As far as replacing the plug that's a good idea if the original looks damaged or corroded. But I would first remove just the brake wire and the trailers battery charge wire from the plug. Then use an extra 15 amp fuse and connect the fuse between the two wires to apply the trailer brakes. Does the fuse immediately blow or do the brakes apply or does nothing happen? It's a quick test that will tell you if you have a wiring short or if the power is even getting to the brakes.
If you have a compass you can have a helper apply the manual override on the brake controller while you hold the compass near each wheel. If the brake magnets are energized the compass needle should swing to point at that wheel.