If you're using WordPress (or software with a similar configuration system), you'd need to set the "Site URL" value on the configuration panel to your new domain. Unfortunately, WP only allows you to set a single domain, so anyone visiting with the old URL will be redirected to your new domain. (Maybe that's not much of a loss with an x10Hosting-provided subdomain, but if you ever plan to use more than one custom domain, you need to install WP as a multi-site.)
As for the cPanel access, you're best off starting a new thread requesting unmetered space, since it's very unlikely that the people who can do anything about it will notice you've mentioned a space problem in this thread.