We've tried everything we can think of. Thanks, Katrina Dear Katrina, It sounds like you are fighting a tough battle indeed. Unfortunately the first step that I can think of is not very frugal but may be required. That would be to put a layer of top soil over the sandy soil. Then I would plant a grass mixture that is heavy with white clover.
Clover produces long roots.
If you don't mind having clover in your lawn instead of regular grass the long roots will possibly help the clover and the rest of the lawn to settle. I personally like clover and think it looks nice. Plus it doesn't mind too much if you tread on it. If this works you may wind up with a lawn that is not traditional looking, but it would definitely be better than sand. Keeping the kids and the dogs off it while it is germinating is definitely something that you will need to do but unfortunately until it is very well established you may have a problem with their activities killing the new plants.