Not very relevant, but how do I replace my old router with a new one?

By reading the user manual that came w/your router, contacting the manufacturer and/or visit their customer support/troubleshooting website.

Turn off the modem and your computer. Remove the cables from the old router and connect them to the proper connections on the new router. Turn on the modem and wait until the status lights indicate communication with the ISP. Then turn on the new router and wait until the status lights show connection with the modem. Lastly, turn on the computer. If all the status lights are green you are good to go.

In most installations, there is NO intervention necessary at any wall circuits. Here's what I do:

1. Go into old router's configuration page, and take screen shots or notes the settings in every panel you customized.
2. Totally unplug old router.
3. Take all the network cables that were plugged into the old router, and plug them into the same outlets in the new router.
4. Plug in the power for the new router.
5. Immediately go into the configuration page of the new router, and refer to your notes for the old router as you set it using the same settings (for router admin password, wifi password, any IP or MAC address filters, any custom firewall settings...)

Now as I said, that's how it works in most homes and businesses. If Cox has some special thing that they do which requires that they go out and flip some switches, that is something you need to ask them about and get the correct answer from them. It is certainly not typical or common for that to be required.

Most people can simply walk over to the store, buy a new router, take it home and plug it in, and set it up. With no intervention from the phone or cable company. It should be almost as DIY-simple as connecting a new TV to your stereo.