This problem is also documented here:
Unable to remove Corporate Sync mail account
Unable to delete failed exchange account w/ hotmail
Corporate Sync Account/Calendar issue - Android Forums
I also refuse to wipe my phone. I've got better things to do with my entire afternoon!!
Any Android developers out there with some ideas?
I had this same problem. I am changing jobs and needed to remove my old work email account. I changed the password on that account and my Droid X could no longer connect. This allowed me to get to those account settings that I could not find before. At the bottom of the settings page is a button for removing account. Good Luck!
Based on the below quote, I found that since I wasn't able to "update my password" or remove my old account, I instead went in and "Added a new account" with the same information as my old account that wouldn't delete. What happened was that no new account ever appeared under Accounts, however, the "old" account that wouldn't remove suddenly started working and I started receiving emails.
It appears that "adding" a new account actually "updates" the old broken existing account if the information matches reasonably. I cannot tell the exact mechanics behind it, but it seemed to work, on my device anyway.
For the record, the account still appears to be "broken" under Accounts, but it at least works now.