Modem/Router Config 150720

"It appears that they are both set up as DHCP servers."

Could you clarify? Both router and modem will give you a DHCP IP, but the one from the modem should be public (IE 68.5.4.3) and the one from the router should be private (192.168.x.x). The modem can't be configured any other way, because it's just acting as a bridge between your router and Cox's systems. As long as the ethernet from the modem is going into the routers Internet or WAN port, and the router's WAN settings are set to DHCP, it should work out of the box. The only thing you have to do is reboot the modem after disconnecting the old router, but before connecting the new router. 

Yes they both should be dhcp servers.  The modem only provides the router with the wan (Internet) ip and the router provides all your devices with internal (lan) addressees.  Your router should be set to obtain its wan address via dhcp.  It should also be setup as a dhcp server for your internal network, that assumes you don't have another dhcp server.