I tried restarting the car and unplugging the CD Changer and plugging it back in again to no effect. However, I noticed that the problem is that the changer's not pulling the CDs out of the magazine ...

My roommate had a Sony 100-disc changer. I believe they make them up to 200 discs. It was pretty good. Once in a while, he'd run into problems where he'd have to take it apart and remove a stuck cd ..