How do I fix an Apple mail account that is "confused"

Question

Last week I moved a mail account from one host provider to godaddy.
I updated the incoming and outgoing mail servers in Apple Mail and did a "Get Mail".
Two new messages came in so it appeared everything was fine, however I realized Mail
was not retrieving any other new messages.  Mail does not report any errors, just does
not retrieve messages.
When I "Get Info" on the account I can see 75 unread messages under "All messages on POP server".
For some reason Mail will not retrieve them.
I can log into the account via webmail and confirmed that the password is correct, and that the account
is working properly.
I have deleted the account in Apple Mail several times and recreated as POP and IMAP with the same
result. 
Apple Mail seems to have "memorized" the account even after I delete it, it acts as if some preference
has incorrect information about which messages have been downloaded?
Any help would be greatly appreciated!
Keith

Answer 1

Thank you for the suggestions BDAqua!
The new Smart Mailbox was unable to retrieve any new messages.
Something strange about Connection Doctor, at the top of the window it reports "Internet Connection Status:  "Mail was unable to connect to the internet, click the "Assist me..."  button for help diagnosing the problem."  (this report is false because the other two accounts work).   When I click Assist me as instructed Network Diagnostics reports "This computer's Internet connection appears to be operating normally. Are you sure you want to use Diagnostics?"
All other status lights are Green within Connection Doctor, it reports "Connection and login to server succeeded" with all three mail accounts and all three smtp servers.
In safe mode, Disk Utilities repaired some permissions, no disk repair was required.
After moving the Cache files, Envelope Index and rebooting, Mail went through the Message Import process but there was no change.
I remvoved MessageRules.plist and backup, still no change.
I'm not getting a password rejection, no rejection or error at all.
In Keychain access I deleted every entry having to do with mail accounts and servers.  I reentered passwords in Mail accounts and still no change.
I'm using no SSL and ports check out.
After all the above I'm still in the same situation, have not found a way to create a new account that is capable of retrieving from my main email box. Very frustrating, I can't seem to figure out a way to bypass the "hidden" library file that's corrupt.
It's curious that the setting in Accounts>Advanced>"Account Directory" there is a "choose" button but you can't click it.  I was hoping I could chose a new location to fix the problem.
Thanks for all your suggestions,
Keith