I tested the mics, which I hadn't used in quite awhile. The capsules didn't seem to be loose, but just in case, I screwed each one off and back on, and got no signal at all in the right channel.
I also tried swapping the capsules. After the swap I still got no signal in the right channel, but the left was fine. So the problem is definitely not in the capsule, but either in the mic base (which seems unlikely), or more likely in the cable or connections.
So let me know if anyone reading does this kind of repair or has a recommendation for someone who does. If this works be better placed in a different subforum, please tell me where would be best. Thanks!