New Site, New Server, change DNS

I searched for a similar problem, but could not find one.

The current site hosted elsewhere, but I now own the domain name at Godaddy. I am in the process of creating a new site via WordPress that will take over for the old site at the same domain name.

I am testing it under another domain I own. I would like to do the following:
1. Install/transfer the new WordPress site to proper the domain name.
2. Change the DNS to read from the new server.

I have moved the files to the domain name…but I can’t do anything with them since the DNS is still pointing to the old server and the old site.

Assuming I did everything correctly, should I be able to edit the WordPress site once the DNS change is made? Or is there another process I need to follow?
Thanks for any help you can provide!