This is primarily a modem issue. The modem has associated with your desktop, so it "remembers" the desktop and it will not deliver a valid IP address to any other device to which it connects, like the AirPort Express.
So the trick is make the modem "forget" the desktop, so that it will associate correctly with a new device......your AirPort Express.
To do that, the modem must be completely reset back to its default settings. That means.....
1) Powering off the modem completely by pulling the power cable from the back of the device or from the wall AC jack.
2) Removing the backup battery from the modem
3) Disconnecting the Co-Ax cable and Ethernet cable
Instructions to remove the battery are here: ARRIS : Touchstone E-MTA TM822
Leave the modem powered off at least 30 minutes, longer is better.
Next, the AirPort Express must be reset back to factory default settings so that it will work with the new modem. To do that.....
1) Power off the AirPort Express and wait a minute or two
2) Hold in the reset button first, and keep holding it in for an additional 10-12 seconds while you simultaneously plug the power back into the Express
3) Release the reset button after the hold period, and allow a full minute for the Express to restart to the slow, blinking amber light
Now power off the Express again.
After you have waited at least 30 minutes with the modem powered off and the battery removed, reinstall the battery, connect the Co-ax cable, and connect the Ethernet cable from the modem to the WAN "O" port on the AirPort Express.
Power up the modem and let the modem run for 10 minutes by itself, then power up the AirPort Express and let it run for a few minutes
Now re-configure the Express using AirPort Utility on the Mac. To do that:
Open Macintosh HD > Applications > Utilities > AirPort Utility
Click on Other Wifi Devices
Click on AirPort Express
The utility will walk you through the setup. You can use the same wireless network name and password that you used before, or set up a new wireless network name and password.