Additionally, make sure to re-sync your controller by holding down the sync buttons located on the controller and Xbox One for a few seconds. For help with this, or other solutions, try the Controller Won't Connect to Console section on the Xbox One Controller Troubleshooting Page.
Very strange, Devean. That sounds more like a software problem than a hardware problem, if I had to guess. If you haven't already, maybe check to make sure that your control setup is correct.
Alternatively, see if you can find any other games that your RT doesn't work for, and see what they have in common with GTA - it might be that the trigger isn't registering as fully depressed, or only ever registering when fully depressed.
Assuming that nothing beyond some solder came off with the wires, you can try resoldering the motor's wires back into place. Make sure to check exactly where they are meant to connect, and not to accidentally solder anything else on the board.
Considering the placement of wires attached by solder on both sides of the top board, it would be quite difficult to reach anything beneath it otherwise. I suppose it depends on what you're trying to achieve.
In any case, you might as well try to move it without desoldering if desoldering is going to be an issue. Make sure not to pull hard, though.
Unfortunately, beyond the parts list given on the main page, I don't know of any other reliable links. I'd suggest Googling the part, or maybe searching on eBay.