Whoops! I totally didn't mean to ignore your comment all this time; I just lose track of emails really easily when I'm only getting them on my phone and the stupid Russian spam isn't being filtered out. /o\

I had already tried to verify my keychain and that didn't help, and then I reset my entire keychain and that didn't help either, and then when I went just now to see if there were any rogue passwords to delete, I didn't see a password for the Mail application at all. Which is what made me think that maybe I should verify my entire disk, and that found major errors in my sector numbers that I had to boot from the recovery disk to repair, and since I booted back up it hasn't asked me for my password once! I think it may actually be fixed! *knocks on wood*