Change Domain but keep Host?

I need to change my WordPress URL to a new domain but keep everything else (all content and wordpress needs to stay/run on the old host – via GoDaddy). I don’t want to reinstall or rehost any files. I’ve bought the domain and have access to the setting the nameserver for both the old and new domain.

Is it as simple as changing Settings > WordPress Address URL and > Site Address (URL) ? Do I need to point the domain to the correct nameserver in GoDaddy preferences?

Any help would be greatly appreciated. I’ve read countless support docs but still can’t figure out this simple problem (ultimate novice here).