Just curious, didn't the old lighting fixtures have two wires running to them (positive & negative)? You should be able to reuse those wires.
If you are using LEDs, you may have them wired backwards (LEDs won't light, if you have the wires reversed). You might try reversing the wires...
Also, some older SOBs (Some Other Brand) trailers use the trailer ball as the ground return for running lights. Not sure if this applies to your Airstream. However, it wouldn't hurt to make sure that you have your hitch is hooked up, especially if your other running lights don't work either.
Alternately, you can temporarily hardwire a jumper wire from the frame of your tow vehicle to the tongue of your Airstream. For testing purposes, a battery jumper cable would work, or a short wire with alligator clips.
One last thing to check: Verify that the LED running lights that you are installing still work by applying 12 volts directly to the leads. In troubleshooting (and perhaps wiring them backwards), they may have been damaged and won't work anymore.