If a defective breakaway switch was applying voltage to the brakes with the pin in place, then you would have full voltage to the brakes (or at least 3 of them in your case) and several other things would have happened in 13k miles towing. You would have a very difficult time pulling the trailer, your fuel mileage would be less than 1 mpg, the plastic breakaway switch would have melted in about 1 hour, the wheel bearings would have overheated and failed from too much brake heat and tire wear would be terrible. Oh, while stored w/o shore power, your battery would be dead in about one day.
You didn't say when you did the compass test if the truck was connected or not? But maybe you didn't do the compass test correctly? You could jack up each wheel and see how hard it is to turn by hand. It is possible that the 3 wheels had the brakes adjusted too tight. Also, maybe you are riding the brake pedal and slightly applying the brakes while towing. And there are probably other potential causes. The 4th brake is ok most likely because it actually has an open circuit someplace and is NEVER working.