You can only use 1 DSL modem on the phone line that comes into your house. So it sounds like you have 2 DSL modems. One Wireless One Not? Hence when the router is online the x-box cannot and vice versa. What I use on my son's xbox 360 is a ethernet to wireless adapter (a cheap 11G - 11 Mbps Linksys one). X-box do an official wireless adapter but its more expensive than the DIY option.
You say wireless may not be the option as reception is poor. We can help diagnosis of wireless signal if that helps?
Or you can invest in a pair of Powerline Adapters.
http://www.amazon.co.uk/Netgear-XEB1004-85Mbps-Powerline-Ethernet/dp/B00167D96S/ref=pd_cp_ce_2
They use the power sockets in your house and provide a fixed connection from one socket to another. You could plug one in near to the X-box and the other one near to the Verizon DSL router. You then plug the ethernet into the X-Box at one end and the ethernet into Ports 1 to 4 on the back of the Verizon DSL router. I am guessing you use one of these ports already for the Desktop PC.
Some people on this forum hate Powerline Adapters as they have issues but I have used thenm without problems for 3 years. Keep the receipts maybe!
