Changing the domain name without cancelling your service isn't possible. However, you can register a new domain name by ordering a new service. Then you can choose to keep or close your current account.

Note: It's not possible to swap domains associated with your Turbify service. The domain that you want your website and email under needs to be signed up for the appropriate plan. If you're looking to forward a secondary domain to your website, you can do this from within your Domain Control Panel.