The solution for us was to change the way that email was sent through the hosted Exchange server we use. We host at Intermedia and apparently their implementation of Exchange doesn't behave when CRM uses the "User Creds" option in the email router to send outgoing email. We created an account that has "Send As" permissions on all users' mailboxes and the email router uses a single set of creds to send outgoing email. Incoming email still uses "User Creds" tho.
Result: No more drafts emails, no more duplicate emails.