Changing Domain Name for Site that already has two domain names

Ugh – I know this question is asked an unbelievable amount of times but it’s so tricky and we aren’t using a standard configuration.

We already have two domain names:
https://www.ouroriginalprimarydomain.com/ and
https://www.asecondarydomainname.com/

All works fine. However, the site itself is not in the root directory. It’s in https://www.ouroriginalprimarydomain.com/test (original test site and when we went live we simply added an index file to the root of https://www.ouroriginalprimarydomain.com (primary domain) to requires WP to load/open require(‘./test/wp-blog-header.php’);

Now the company has literally changed the name of the company and has registered a new domain: https://www.newcompanydomainname.com/

Using godaddy control panel, we pointed https://www.newcompanydomainname.com/ so that if that url is typed in, the site loads. Of course, once anyone clicks a link, the url reverts to the origina lhttps://www.ouroriginalprimarydomain.com (which is really loading https://www.ouroriginalprimarydomain.com/test).

So….we need to change both the primary domain name (so that https://www.newcompanydomainname.com/ is primary) and we need to change wordpress site address and url address so that it uses https://www.newcompanydomainname.com/

I am terrified that once I change the domain name in the dashboard, and then change which domain name is primary in my godaddy control panel, that everything will be completely messed up. I think this because I tested this on another install on another server and the entire site was whacked, so obviously I did something wrong – but I have no idea what I did wrong.

Note, I am not “moving” files or directories. The set up is fine the way it is. I just want to change the primary domain name in godaddy and change the wordpress address and site url. Any assistance you can share would be very appreciated (assistance as in, what order do make the changes? And, judging from the fact that when I tried doing this on a test install and the site was completely whacked, any ideas, based on that result, as to what I may have done wrong?)

Thank you SO much!