It works! Or rather it doesn't work which is good!
Just powered on the car after replacing the license plate bulbs and they don't go on when the lights are off and they go on when the lights are on. I guess one or both of the old bulbs created a short? I doubt that removing and installing the LCM fixed it. I guess I won't be needing another LCM after all. The only other things I replaced were the front door light switches. The dirver side one was really bad but decided to replace the passenger one as well. They're cheap and only take 30 seconds to replace wihtout any tools.