There has been a long standing bug in Mozilla Thunderbird such that drafts saved with Thunderbird do not correctly open up for editing in the FastMail web interface. This is a confirmed bug with Thunderbird when saving drafts to IMAP servers.
Here's a link to the existing bugzilla report. If you have a login at the bugzilla system, please vote on the bug to try and increase the chance of it being fixed. The duplicate bug even pinpoints around where in the code the problem occurs, so hopefully it shouldn't be too hard to fix.
While we're at it, if anyone can badger a thunderbird developer to start looking into this horrible bug (and partial fix) with downloading large IMAP messages with attachments, that would be great as well!