How can I resolve my VPN and DNS issues?

Depending on how your VPN is configured, you might or might not use the same DNS for your VPN and for Internet. VPN's are (typically) like an additional IP stack on your system, and can have a separate DNS server address configured. But not all systems do this.

If your VPN does not assign a new DNS for the VPN session then you will continue to use the DNS server(s) configured in your main Internet IP Stack. This can present a problem if the external DNS cannot resolve internal addresses (or as you indicate, if you don't want internal addresses to be known externally).

If your VPN does assign a new DNS - for example by using DHCP option 6 "DNS Server" - then you can have different DNS servers for the VPN and for Internet. Your OS must support this, as must the VPN service. If you send traffic out both stacks at once this would be "Split Mode".

A final option is that you might operate your VPN in Tunnel Mode, sending all communications (including Internet) through the VPN stack. In this case, when you are on the VPN and DNS, use the VPN's DNS. This is probably the most secure way since all internal traffic is sure to stay in the VPN.