name servers

ask your new host to migrate the your files over using the WHM transfer options, once everything is done, login to your domain registrar to update the domain name server for your domain to point to the new web host DNS server.

 

It sounds like your host is in control of your domain name, so you are going to have to ask them to change your nameservers. Providing they are a good provider they will do this without question. Its not uncommon to find some who will have a go at you though when you ask for your domain details to be changed.

On the other hand if you have control of your domain you can update nameservers yourself, your new host can migrate your account or you can create backups and then restore them.