Dust control for roads is commonly accomplished by either spraying water or by adding calcium (or magnesium) chloride (which absorbs water and makes the road moist). Polyvinyl alcohol is not as good. I would reconsider asphalt. The only serious solution is to use cement or asphalt, and asphalt is cheaper.
As for removing sand and clay, the only option is physical removal. A shaker-screener works for wheel barrow loads, but I suppose that's not helpful. Industrial level screening is an answer, but it's also absurd.
I recently saw another possible option... mulch: Can I use mulch as a driveway covering? which might be of interest. Perhaps you could try it in an area to see how it does (or doesn't) affect the dust. For the price, it would be worth testing.
And I came across a patent where a mix of 5-10% gypsum and 5-10% gluten in water can been used to control dusty roads too. I'm sure there are lots of other dust controlling concoctions but calcium chloride or just water will be the cheapest.
edit-
renting a bobcat or getting a front loader for the tractor would be smart. Just dig out the old stuff and lay down some new/good gravel.
Read about asphalt emulsion.