Your DHCP server needs to pass out the IP address of your DNS server (only).
In general, you should not include any references your ISP DNS servers anywhere on your network. Not in your DHCP server, not in your clients, and not as a forwarder within your own DNS server configuration.
When you make the change, renew the leases of the devices that have the old address if needed, and alter the hard-set DNS server settings in the devices that don't use DHCP.
The one sort-of exception to this: your DNS server should have its Network settings DNS server address set as 127.0.0.1; as the localhost address. Not as its IP address.
And in general, routers don't do DNS. (Though it's common for folks to refer to devices that provide routing, NAT firewall, potentially VPN servers and other services as a "router", technically routers don't typically include DNS services. And very few gateway devices - even the fancy ones and expensive ones - implement DNS services.) Routers (and particularly the usual sorts of network gateways) can generallty have a DNS configuration for two reasons; to forward DNS requests to an upstream DNS server(s), and to have a DNS server address for an embedded DHCP server to pass out to DHCP clients.
In your particular case, the IP network connectivity is very likely present in both of your test configurations, it's that your clients aren't getting DNS translations.
Probably more than you ever cared to learn about setting up DNS services on OS X Server.
