To convert an addon domain from one cPanel to another cPanel:

Download the content including any associated databases of your addon domain to your local PC.
Go into the original cPanel and remove the addon domain from under the Addon Domains icon.
(You do not need to delete the original content or original databases.)

Create a new account inside your WHM. Fill out the form, using the old addon domain name as the domain. Click the Create button.

Once the new cPanel is ready, upload the content of your old addon domain to this new cPanel, in the public_html folder.
(When using FTP to upload, connect to the cPanel's IP address and login with the new cPanel's username and password.)
Log into the new cPanel and restore the database backup(s).

Reconfigure any database connections.This ensures that the database name and username being correct regardless if having been changed.

Your site will only be down as long as it takes to complete these steps. There is no need to change your name servers.