Assuming you are talking about a mailbox being hosted by DreamHost, you don't have to do anything but change the e-mail address. Find the mailbox In the Account Control Panel under "Manage Email" and click on "edit". Change the email address and then click on the "Edit Address" button.
Once the changes take affect, the old address will stop working and messages to the new address will be delivered to the mailbox.
As far as transfering messages from one mailbox to another, you can do this using an IMAP client like Mozilla Thunderbird and just drag and drop the messages between folders. Be aware that means the client downloads the message and uploads it again, so it may take a while to complete. But that will work with both FTP/shell and mail-only mailboxes.
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