After struggling through the three different methods for changing my domain name (http://codex.wordpress.org/Changing_The_Site_URL) so that my existing blog/site can be browsed to, using a new (better) domain name, Hostgator rescued me with a really simple way to do this:
1. If you have the new domain name as an add-on domain, remove it from add-on domains
2. Change your primary domain name to the new domain name at https://gbclient.hostgator.com (you go to "View Hosting packages", and there is a "Change my domain" command at the top).
Done. It also changes automatically to the new domain name in the Dashboard>Settings>General, while preserving your dashboard password.