When you say 'you have litchick.net', I assume you mean you registered this domain as part of the sign-up process.
You can remove the hosting associated with litchick.net, but you can't actually delete the domain registration, you just don't renew the registration when it comes up for renewal and let the domain registration 'lapse'.
You can register the new domain you are interested in via the panel at Domains -> Registrations. Note, this will cost you $9.95 for 1 years registration.
Once you have registered the new domain and received the confirmation email, you can add it to the hosting system in the panel at Domains -> Manage Domains -> Add New Domain/Sub-Domain. After doing this, you will have another web-directory in your home directory, by default this will be named the same as your new domain (ie: yourdomain.com), this is where you place your files for the new site.
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