I want my old domain name to keep opening my website under the new domain
If you would like to have another domain opening the same content as your primary domain, you should add the old domain as „parked“ for your account. This could be easily done through your cPanel -> Parked Domains.
To add a domain as parked its name servers should be pointed to the server which hosts your account. You can find the correct Name Servers for the account in your Welcome Email that you received upon sing up