Moving server to new location and resulting new IPs

Changing the server's primary IP address is done in the operating system's network configuration scripts, not in cPanel. Once that is done, use Home » Server Configuration » Basic cPanel & WHM Setup and change the main IP address there, so that it will be written in all of the cPanel configuration files.

Then, use Home » IP Functions » IP Migration Wizard. It is specifically designed for situations like this. It will explain all of the changes it will make, before it makes them. You can read its documentation here:

http://documentation.cpanel.net/display/ALD/IP+Migration+Wizard

Also, remember to register your new IP addresses as nameservers at your domain registrar, so that your nameservers will work. Unfortunately, there is no way to completely avoid all DNS propagation delay, because of the way nameservers cache queries around the globe, so it may be a few days before your sites resolve to the new IP addresses globally.