Hi HappyHealthy365.
It does look like your domain didn’t get registered.
I’ve come across issues like this in the past where I’ve bought a domain that I Couldn’t believe was actually available. To find out that the Domain Provider had some issues and that it was actually already registered. In your case the domain isn’t registered at all, So I’d go check with GoDaddy to find out whats going on if you’ve already paid for it. They may have had some issues at the time and couldn’t process it.
Also be sure to change your Nameservers to those provided by HostGator.
That will make your domain point to the Hosting account, In the meantime if you have your IP address to the server that you are on, You can use something like this to view your website and make edits.
192.168.1.1/~accountname (Where “accountname” replace with the your username to login to the Hosting Account) This whole link will be your URL to your Website, To Access wp-admin. Use something like this. 192.168.1.1/~accountname/wp-admin/
Good Luck! ??