If your host/registrar won't let you transfer a domain name to a different registrar-- or, conversely, change hosting while keeping the original registrar-- you've got the wrong host/registrar. The "one name included free" is just a sales point for hosting plans in a certain price range. It shouldn't be a restriction.
A domain should be purchased from a reliable domain registrar. You can try what domain extension you need and if it available or not. Most of Top Level Domains are already purchased or running. So you can try your domain name for .com, .org, .biz, or other domain extension. Read terms and conditions prior to purchase a domain. Also ask for its hosting. Hosting of the domain is also a crucial factor to run a website smoothly without any problem.