I have done as you have suggested; only time will tell if it is "the magic" I've been looking for.

I shut down, sent myself a message, restarted, and waited for a new message notice (the webmail notifier told me immediately that there was a new message). I got the notice a minute or three later, read it, closed the e-mail, sent myself another e-mail, got the notice, opened the e-mail... so now I really just have to wait to see if I get any more errors.

I think I've found the problem with messages getting marked "read" or not:

When I opened an e-mail from the notification box (bottom left of the screen), everything opened correctly, but the message was not marked "read" anywhere else. When I read messages in the Modest app itself, the e-mails are properly marked "read" everywhere.