Download all of your website files and upload in new cpanel files manager ,....Next download your old database . Create new database in new cpanel account... assign user to it ..., Later upload the downloaded database to new database created. Update configuration.php or settings.php or config.php of any web application( let it be wordpress, magento, joomla, etc ) add the new database name, username, password.
Once you done the things as above, Later update DNS records in your domain registrar panel..