Hello,
The first line is a bit confusing, it implies that you want to change the DNS zone name. Is that the purpose or are you just moving the DNS zones to a new server?
If the DNS is AD integrated, and you are just replacing the DNS server, just make sure your DHCP hands our the new DNS server IP and you change any static IP's to the new server's DNS. You can also add the old DNS server's IP address as a secondary address to the new DNS server's NIC.
If you are changing the Zone name, that's a whole other issue that will be more complicated because your ADDS services depends on the zone. You would be in essence changing the domain name.
Please clarify and forgive me if I misunderstood the question.
Cheers,
Miguel Fra
Falcon IT Services
https://www.falconitservices.com