If you are moving a domain from one Pantheon site to another, you do not need to register for CalNet or fill out the Offsite Hosting form again. You will need to fill out the Upgrade to Paid Plan form to choose a hosting plan for the new site.
Once you have received confirmation that the new site has been upgraded to a paid plan, simply remove the domain on the old site's dashboard and add it to the new site's dashboard. The DNS refreshes on demand, but there may be an unpredictable amount of delay before the domain is released to be added to the new site.
Once you have moved the domain(s) to the new dashboard, if you no longer need to use a paid plan on the old site, you will need to fill out the Update Plan/Stop Billing form to stop billing for your site. If you do not fill out these form, you will continue to be billed for the old site.