Hi Sandeep Goyal,
You can continue using your existing custom domain name and also include the private dns zone for postgress dns that is required in the private endpoint configuration so the configuration will looks like:
- xyz.com (Your custom domain) --> ips_vms_vms
- privatelink.postgres.cosmos.azure.com(Postgress domain) --> ip_pe_postgress
https://learn.microsoft.com/en-us/azure/private-link/private-endpoint-dns
About "single VNET can have only one private DNS zone." the documentation refeer to the creation of the link itself However the vnet can be associated to multiple private dns zone by the private dns zone link.
You can create only one link between a private DNS zone and a virtual network.https://learn.microsoft.com/en-us/azure/dns/private-dns-virtual-network-links
So you can continue using your exiting domain for you application and also add the private dns zone configuration as is required by private endpoint configuration on postgres. Let me know if you have more doubts,
Luis