The Cox network bases it's DHCP leases on your MAC address. If, in your case, you had your computer directly connected to the modem and it was working fine, then you connected the router and nothing, it means that the Cox servers don't recognize the MAC address on the router's WAN port. Just reboot the cable modem (making sure to leave the router on, toherwise the cable modem can't see the MAC) and be sure that everything on the router is set to DHCP (at least the IP, subnet mask, and default gateway.)
I've been using cox for a while and this is the joyful procedure I normally get to go through when I replace any of my border routers.
As an alternative, if your router supports it, just have it spoof/clone the MAC address from your computer's NIC. This will make the Cox network think it's the same computer connected, and allow it to continue with the same settings.
As another note... do *not* set the IP, subnet or gateway to be static. Cox actively scans the network for these configurations and they usually don't stay online for more than a day.