| How can I make sure both xyz.org domains are properly setup? |
I think you have made a mistake, You cannot have two domains with the same name accessible on the same network
If this xyz.org domain is a LAN domain with no one ever needing to connect from outside the LAN then there was no need to register it. You can call your LAN domain anything you want. Registration is only required is there is to be public access.
The fact that you purchased a domain name means very little. It does nothing unless there is a DNS record attached to it. That happens when you have something for it to connect to. As this is an internal LAN domain with no public access then you need to set up the DNS record yourself.