If you have DNS service paid for and can use it seperately from the rest of thePlanet services, you should be able to use thePlanet DNS to point to your new servers.
At any rate, you will need to set up new servers first, get their IP addresses and then either: a) point thePlanet DNS service to new IPs; or b) order new hosting provider to create DNS entries for your new IPs; and ask thePlanet to close their DNS service.
Note that DNS is a bit tricky in that changes tend to take time to propagate (could be a day), and even more time, if you change the actual DNS service provider.
During the transition phase there may be hosts on Internet who will resolve to your old servers and other hosts who will get the new IPs. Therefore you should also set up redirection on your old servers for some time.
Precisely for these reasons, I personally tend to use DNS service completely seperately from the rest of hosting, and change it as rarely as possible.