Hi Mike,
[[MacBook Air Troubleshooting|This troubleshooting wiki]] might provide some guidance on how to go about finding the problem. It's the only one I could find for MacBook Airs, linked from the A1237 and A1304 models, but the causes, effects, and and repairs just might hold true. If nothing else, it's a place to start.
[https://www.ifixit.com/Story/21244/How_I_avoided_throwing_away_a_%241200_MacBook_Air_%282013%29|This story] deals with a different issue, but it might give you some inspiration-power to get through the troubleshooting and repair process! :)