The best method to move MySQL databases is to export them from your PHPMyAdmin tool. If this is not already installed, you can do so through Site Manager >> Install & Manage. The tool is accessible with your Site Manager user name and password through http://yourdomain.com/pma/
You will click on the database name and then utilize the default options in the "Export" tab.
Once exported and thus backed up on your local machine, you can set up the databases and associated user name/password combination through cPanel >> MySQL Database Wizard. You will need to go through the wizard for EACH database that you need to set up. Do not lose the database user name and password that you set up for each database!
Once that is done you can connect to cPanel >> PHPMyAdmin [it is installed by default on all cPanel servers] and navigate to each database in order and utilize the "Import" tab to upload the backup files for each database.
If you run into any downloading or uploading trouble with your databases, please contact support!