Changing Domain Name from Secondary to Primary

I’ve searched the forum for a solution to this, but my situation is a little different, and I want to be SURE I have the right instructions (I’ve already rebuilt the site once and don’t want to do it again!).

Here’s the situation – I built a WP site that is hosted at Newtek. The domain for the hosted account is ourhousedesign.com, and that is the domain that is referenced throughout the WP settings and database. Another domain, lynnadamo.com, points to the site. The client wants to change the “primary” domain to lynnadamo.com, and have ourhousedesign.com be the “secondary” domain (and eventually retire it).

This will require WP changes, database changes, AND changes with the host company. I want to know what I need to do, and in what order (i.e. make the WP changes first and then swap the domains, or vice versa)? Your help is much appreciated!