Most domestic cable ISPs have the cable modem configured to recognize only one client computer by the MAC address of its network interface.
Once the cable modem has learn the MAC address of the first computer that talks to it, it will not respond to another MAC address in any way.
Thus if you swap one PC for another PC, the new PC will not work with the cable modem, because the new PC has a different MAC address to the first PC.
“I don’t like wireless so that’s out of the question”
Seeing that you do not want wireless using a network switch would be the simple way to go
D-Link 5 Port 10/100 Unmanaged Ethernet Switch
How to connect the switch to the Modem and Devolo Units
You need to power off the Cable Modem, Switch, and the Computer,
Connect the Cable modem to the Switch using an Ethernet cable to port 5 on the Switch
Connect a second Ethernet cable from port 1 on the switch to the LAN card on the main computer.
Plug in the first Devolo to the electrical outlet and connect an Ethernet cable to port 2 on the switch
Plug in the second Devolo to the electrical outlet in the room that require access and connect an Ethernet cable from the Devolo unit to the second computer
Wait 30 minutes
Power on the cable modem, and once it has rebooted and gone fully online again all indicator lights settled down,
Power on the switch and wait 1 minute
Power on both computer and wait until they are fully rebooted
You should now able to access the Internet on both computers
How to effectively set up file and printer shares if needed
Go to this site and take a read and follow the instructions