If the old server on Fedora is working, just leave it alone on the public internet.
Copy everything you need onto the new server and only have that server visible from within the private LAN.
For dev/testing purposes on the new server, just add eg _dev to the domain names and only have those available on the internal LAN DNS.
In fact, better is to only add them to your personal machine hosts file; that way no-one else can play with it whilst you are dev/testing.
DNS is not required in this case.
The public DNS is likely hosted elsewhere eg your ISP and will be updated regularly (eg daily ... hint) by them.
Incidentally, as this is for a work server, make sure you sue the LTS version of Ubuntu.
Note that Fedora was a really bad choice; that's the RH bleeding edge R&D distro; definitely not recommended for a work server.
Better would have been Centos (free version of RHEL).
You'll find this site full of free to read manuals/books www.linuxtopia.org