First, the CGM4981 is a modem/router combo(AKA gateway) not a stand alone modem. That is what the G stands for. Do you pay for it or get it for free with your contract? If you pay for it, return it and get your own stand alone modem. I suggest the Arris S34 but see here for full list. Look under the section labeled "2Gbps or less" and where integrated wifi is set to "no". The Technicolor CVA4004 is another option and is sold/supported by Cox so you can swap it at a Cox store if you need to.
If you get the gateway for free, is it in bridge mode? Also when you have the problem, what speed do you get direct to the gateway? Remember, in bridge mode, every time you change what the ethernet is connected to, you have to reboot the gateway.
Yes, I had put it in Bridge Mode - otherwise it would not have been working before it stopped.
Yes, I reboot the gateway when trying to get things working again.
It will work with another router in router mode. Some people purposely do this to separate networks for privacy reasons like if two roommates are living together. It just isn't optimal performance wise.
