Your router wouldn't get a public IP address. Only a provider (Cox, Verizon, AT&T, etc) assigns public IP addresses and this 1 public address would be assigned to your modem. This 1 IP address is how the rest of the Internet "sees" you.
You don't want the Internet seeing your personal router. Your personal router manages all your private IP addresses. The Internet doesn't see these addresses...nor the private address of your router.
Only the Cox-side of your modem gets the public address. Plugging a computer directly to the WAN port of a modem wouldn't involve another IP address because this connection is only at the MAC layer. It's a NIC-to-NIC connection so it only involves MAC addresses.
It sounds like you're having IP conflicts and when this happens, routers can terminate communications until its resolved.
Your routers may be issuing private addresses without issue but do these devices roam around the house from WiFi to WiFi?