The modem doesn't issue an IP address to your separate router. When the modem establishes connectivity to Cox, Cox issues a public IP address to your modem. That's what the world "sees" you as. Your router has its own private IP address and issues more private IP addresses to your devices. The modem and router...albeit different networks...only agree to pass data.
Besides the IP address, what was the problem when the modem linked correctly to Cox...no Internet service? If your devices could connect to the Internet, the first modem was good.
I suspect it's your WAN service. You'll need to post some speed tests and modem logs to verify.
So I'm not expert on cable modems (pretty good on more 'standard' corporate networking)... whatever device is providing DHCP to my router (assumed the modem was doing that) is not doing so properly. totally get the public private IP differences. What happens is My router gets issued the cox IP with a 24 hour lease, but it goes away sometime in in the next few hours. Once it is in this mode, It won't reissue an IP to any device unless the modem is restarted. When it is 'working' e.g. after a fresh router reboot - all works as expected. I can plug in a router or a pc directly and I will get issued a cox IP just fine. I can plug, unplug, change devices... all works great. Additionally as soon as it fails, the link light on the ethernet port on the router goes out in case that is an indicator of something. However, even when it fails like this - the front light indicating internet stays lit and the phone still works. Seems to indicate to me it thinks the link to COX is still working.
to your other question... while I'm connected.. there are no issues. Speed is 180Mb and is the same as I had w prior modem. I can run speed tests from my LAN and those are either rock solid or 0. I've had a tech out - they look at all the channels and levels and say all is good. Do I have access to these modem logs?
I have had a surfboard modem (not sure on model . 6xxx I think) and it was rock solid for many years with all the same gear on the lan side.