You should run DNS on your Domain Controllers and host an Active Directory Integrated DNS zone for your AD there. This should be automatically configured after running
dcpromo and opting to install the DNS Server role (I assume that you already are doing this. It's a basic step to setting up AD).
Then, your Windows clients should be configured to use your internal DNS servers for DNS resolution.
You can set this a few ways. The best way is to configure your internal DHCP server to tell your DHCP clients to use the internal server(s). This can be configured by setting DHCP Option 6 to the IP address(es) of your Domain Controller(s). Please, don't use a hosts file entry for this
If you still wish to use your external resolver for external DNS lookups, just set a global forwarder from your internal AD DNS servers to your external DNS resolver. At no time should your internal domain-joined workstations use any DNS server other than the ones that host your Active Directory zones.