Follow these simple steps:

Old Server Task:

• Login to your old cPanel
• Go to Backup Wizard -> Back Up -> Full Backup
• Click on the “Generate Backup” button (Home Directory)
• Download the backup file to your computer
• Extract the file and go to homedir -> mail -> domain.com (folder). (You will find all Email accounts inside the folder)
• Select everything inside the folder (domain) and compress to upload in the new cPanel account

If backup wizard is hidden then you can follow

https://serverhostname.com:2083/cpsess1268795626/frontend/paper_lantern/backup/wizard.html

New Server Task:

• Login to your new cPanel
• Go to “Email Accounts” and create all email accounts those were in the old cPanel.
• Then go to File Manager -> mail -> domain.com (folder)
• Upload the compressed file here [Old Server Task last point]
• Extract the compressed file and you are done.

Now login to your email accounts and find all your data.