I've tried a couple different controllers, including an N64 emulator controller and a XBox 360 controller, with no success in getting my controller to even be recognized by KOTOR or KOTOR 2. They both downloaded drivers when I plugged them in, the Xbox 360 controller has up-to-date drivers, and it works in the Xbox store. How can I get my XBox 360 controller to work with KOTOR and KOTOR 2?

KOTOR 1 does not have native controller support for PC.

KOTOR 2 does not have native controller support for PC, at GOG. The steam version has a patch with controller support.

I am using this Xpadder profile to play both, but I still find myself resorting to the mouse from time to time:
https://www.youtube.com/watch?v=_7a3XeWhC-4

Use AntiMicroX, it works great and is free.