Had to remove one of my headlights to fix it - it had the brass gear fix, but the bracket retaining the headlight motor gear cover was improperly installed, so it had broken loose. It was jittery while driving, didn't always go up or down all the way, allowed excessive side-to-side play, and was no longer waterproof...

Anyways, this process is fairly well documented on the corvette forum, and the other GM cars of that generation with popup headlights are fixed in much the same way.

There's only one thing missing: the wiring harness for the headlights and the headlight motor is clipped to the car, and it may seem impossible to take it off. That's because underneath, there is a clip, and it looks like this:

To take it off, you can either slide it forward - but mine was stuck, so I had to take the entire clip out. You'll need relatively thin, long needle-nose pliers, in order to push in the sides of the clip as you pull. Someone once posted this image, but it was a dead link, so hopefully it'll be helpful for you.

Key words for google: 1997-2004 (97-04) C5 corvette headlight motor wiring harness stuck or won't come out due to headlight harness retainer clip.