So you can send and recieve mail using a thick client? Just not the web-client interface?
If they have separate names/IP's & your just doing this with a few accounts:
Open thunderbird and drag folders between the 2 servers.
If you don't want to mess with IP's/DNS & still a few accounts:
Power up the old 'box' (or if we're not talking 2 machine here the old /opt/zimbra), copy the mail to a local foler in thunderbird, shut down old box, power up the good server, transfer back the mail.
For a larger number of users-If they have the same IP/DNS entries:
Your going to have to change the ip address assigned to one of the boxes; better yet do the entire imap sync by ip adderss and not names.
Because I don't think you want to be messing with ZmSetServerName right now...
An imapsync example:
User Migration - ZimbraWiki