Mail deception - says messages are sent out but are not delivered

The Mail program appears to send my outgoing e-mail messages but does not actually do so. When I send a new message or reply to a message, Mail behaves as though the message has been sent: the spinner runs and the swoosh sound occurs when the message leaves my Outbox. The message then appears in my Sent folder. However, the mail is never received by the intended recipients. I apparently operated like this for weeks until my sister mentioned my lack of response to her e-mail a couple of weeks prior. (Honest, I really did reply.) Upon investigation I learned that others had not received messages I had sent. I then tried sending a message to myself at the same account and at my work account (different ISP) and the messages were not received.

I am using a POP account at core.com. If I access my account via the core.com webmail site, I am able to actually send messages. Getting them from the Mail program on my computer is the problem.

I noticed several other questions posted about Mail not sending messages but those queries mentioned receiving an error message or the program timed out trying to send and knowing immediately the mail was not sent. My situation seems unique in that the program is behaving deceptively - it tells me a message is sent but it is not.

I followed the steps Apple posted regarding Leopard Mail not sending messages but no success. I am using OS X5.6 on an Intel MacBook.

I have spoken with the core.com technical support staff and they assure me the problems is with the Mail program or my iMac. They are most likely correct. As an experiment I set up MS Entourage as an e-mail client for my core.com account. Entourage had no trouble sending messages to intended recipients.

I have double and triple checked the settings in Mail Preferences. The core.com technical support representative confirmed I am using the correct settings and suggested I use a different e-mail client program and forget about Apple Mail. However, I understand it is not possible to export information from Mail to a different e-mail client program so I am reluctant to make the change. The representative also suggested I remove Mail and reinstall it as a way of possibly correcting the problem. I do not want to go that route if I can avoid doing so. If that is the best approach, how do I go about saving my messages and addresses in Mail before removing Mail? Other suggestions to get Mail to behave properly?

Thank you!