How to change URL without causing issues?

Want to transfer current WP site over to a new domain. Made the bad decision to just change both the WP address and site address from WP admin general settings and couldn’t even log in after that. Had to follow instructions in a Youtube video to go through CPanel file manager and reverse the changes.

Old domain (mericon.com.au) is using Bluehost nameservers, new domain (maisongroup.com.au) is currently using default nameservers as switching them over to Bluehost seemed to disrupt our ability to use Gmail.

Reading support articles is confusing me, as I don’t fully understand the ins and outs of hosting, nameservers, roots, directories etc. Can someone please walk me through the steps just so that our website runs under the new domain without affecting or relying on anything else? We will eventually be closing the old domain.