I took the strings off and took the neck pickup out... and I was right! Not only did the pickup slide down from the pickup cover, but it completely came out and was just sitting in the pickup cavity! I guess someone who thinks they may have a similar problem could diagnose this by shaking the guitar and listening for a pickup banging around inside.
I pushed it back in the cover with a good amount of electrical tape adding extra support, so hopefully it will stay in there and the problem will be all fixed. I'm going to rewire it anyway to see if I can try to clean up the insides a bit.
