We can get a full cpanel backup which includes your email accounts, you will need to change your POP3 and SMTP mail server settings to continue receiving email once the transfer is complete. Please refer back to your "New Account Information" email for server settings. You received several emails when you purchased your new hosting account, try to keep all of them so you can retrieve your login information too.
We can change your DNS settings for you once the website is transferred and you assure us that the website is operating properly.