This time, we took it down and ran checks on the database tables, and other constructive maintenance. Nothing horrible.
After some poking around, we think that it might be the slave database (kenny, which the directory is running off of) crashing the master database (cartman, the one that runs the site...) during replication. Brad turned off replication, but left the directory up.
So, the directory is up, but you can only search on old data. This isn't as big
an issue as shutting it off entirely, however.
We'll keep working, and keep you posted.
[Short update: Sorry I didn't post this sooner, but no, we do not want to switch database software. It's too much of a hassle, or costs too much. Any system we use is going to have its own set of bugs.]