After years of fiddling with lights on various trailers, I finally gave up and normally figure I am going to have to replace connectors at least once a year, when dealing with factory stuff, I have found it much easier, just to un-plug and plug a new harness in. When they started putting computers in cars, it became much more difficult to tap wires because you never know which wire is being monitored by that computer and it often times leads to check engine lights coming on. My buddy tried to work on his and ended up burning the computer up in his Nissan truck, that was a very expensive lesson to learn.
If you have two pins lighting up with the turn signals, you have a short in that system somewhere or you could just have one that was wired wrong from the factory, that does happen, also pins get hooked up wrong in the plug when they are molded, do you have the factory color coded GM wiring diagram for the year you are working on? Make sure you have the correct one, because all of the companies have a tendency to change colors on their wires over the years.
Another thing, I was just checking some of the manuals I have, did you make sure that the maxfuse is installed so the power lug under the hood has power, some of the later model GM vehicles came with a blank fuse installed where the 40 amp maxfuse needs to be installed. Do you know if the previous owner towed with this vehicle?