does the AE need to stay plugged into the modem by ethernet in order to work?
Yes, if you want the AirPort Express to act as wireless router and provide a wireless network.
I want to hard wire my xbox into the airport but it doesnt work when i plug it into the LAN port.
Temporarily, use a laptop or desktop and connect the computer to the LAN port on the AirPort Express using an Ethernet cable. Be sure to turn off the wireless on the computer for this "test".
See if you can get an Internet connection that way on the computer. If you can, then you know that the LAN port is working correctly on the AirPort Express. Check the settings on the Xbox again and reconfigure it if necessary.
If you cannot get an Internet connection during your "test", then try another Ethernet cable that you know is working. If things still don't work with another Ethernet cable, then the LAN port on the AirPort Express may be defective. Exchange the AirPort Express for another.