To migrate your email accounts, follow these steps:
Old Server Task:
Login to your old cPanel account.
Go to Backup Wizard -> Back Up -> Full Backup
Click on the “Generate Backup” button
Download the backup file from File Manager -> home/username
Extract the file and go to homedir -> mail -> domain.com (folder). You will find folders containing all Email IDs. All these folders are actually email accounts containing email data.
Select all folders and compress to upload in the new cPanel account.
New Server Task:
Login to your new cPanel account.
Go to “Email Accounts” and recreate all email accounts those were in your previous cPanel accounts
Go to File Manager -> mail -> domain.com (folder)
Upload your compressed file here from the backup folder [Old Server Task(Point 6)]
Extract the compressed file and you are done.
Now login to your email accounts and find all your data.