I give you a better solution, let's try.
What do you need?
-Microsoft Outlook or any email client
-Your email must be an IMAP account
The idea : sync your emails via email client (IMAP account)
- First, dont touch your current email (I call it email email_A@domain.com hosted at your old mail server IP_1)
- Go to CWP, add your domain, create new email account, of course you will have the inbox email_A@domain.com (exactly the same email address, but hosted at your new server IP_2 )
- Now you launch Outlook, add your new email but instead of input your mail.domain.com (mail server address, it might be different base on your configuration), you input your new IP (IP_2)
- drag all emails and folders from email_1 to email_2 (maybe drag 1 email for a test)
- Wait for the email to be synced to new server (actually it just upload)
- Go to roundcube on the new server, verify it
- If everything looks good, you can continue to sync and disconnect your old account in Outlook