I believe its the SB6131 or SB6141. Terribly so, I am traveling for work...which is when this typically happens.
Yes, If plug in a computer I will pull the typical address and then if I plug the new router in and reboot it will pull that address correctly. However, only until the cycle begins again next DHCP lease expiration.
DHCP does not come from the Modem, it only advertises it to the DHCP server and proxies OFFERS, REQUESTS, and ACKS, which is not commonly on the same network subnet as the user end devices. ARP Caches have nothing to do with this typically, as DOCSIS is a layer 2 technology and the IP to MAC mapping doesn't apply when you don't have an address yet. If there WAS an ARP Cache issue it would be because the next router in the path is not purging ARP entires when they expire or is otherwise misconfigured (like enabled Proxy ARP, etc.)
I've scoured several other forums regarding this and a lot of commercial routers have this issue with Cox (by name). If you wait long enough on hold or get the just lucky tech who understands that there is possibly a DHCP FLUSH issue on the server side. This is further evident by the fact that I cannot resolve this issue by Spoofing the MAC of a computer or secondary device.
I believe Cox might have misunderstood me, when I wrote this thinking i was merely venting, but I actually would prefer them to solve and fix this issue. This has been ongoing for several years for other customers and instead of resolution they often move to another Company. Guess what CenturyLink offers 1Gb in my neighborhood for $20 less than what I am paying for non-working service.