Good morning and thank you for your response. Yes, I am plugging the ethernet cable from my cable modem into the WAN port on the AEBS. My Airport is the original version that looks like a flying saucer, but the connection ports are basically the same. After doing this, I have a steady power light (center) and the lights on both sides of the center light are constantly flashing.
OK. Leave the Ethernet cable connected between the AEBS and the modem. Pull the power to both the AEBS and the modem. Leave the power off to the modem for 30 seconds. Plug the power back in.
Now, we're going to do a factory reset on the AEBS. Straighten a paperclip and press the reset button (which is recessed into a hole). Continue holding and plug the power in. Wait about 5 seconds until you see the three status lights glow, then release.
Allow the AEBS about 30 seconds to initialize, then test again. If this doesn't do the trick, you may need to contact Cox and ask them if they require a special configuration for wireless routers. For example, you may need to duplicate the MAC address of your computer if it's specifically provisioned in the modem (I doubt this is the case as most ISPs have move away from this strategy as wireless routers have become so common).
The only other possibility I can think of would be that they use static IP for client devices, but if that were true, you would likely be aware of it.
And of course, there's the possibility that you've just got a bad AEBS. I hope that's not true, but it happens.