If you do a nslookup in local machine where the dns server is being run at, of course it would resolve, locally.
If you register the domain name at Godaddy, you also need to register your nameservers with Godaddy. This does not mean you need to purchase the nameserver domain name. You can simply login to your domain control panel at Godaddy, and register your nameservers for free. Read their documentation for more information.
Once it is registered, you can use the nameservers for all your domain name.
Check your nameservers to see if it's registered at: