How do I make my cheese "crackers" less greasy?
I'd first try experimenting with different cheeses; different brands or ages of cheese may behave differently.  You can also try hard cheeses, as I've made something similar with hard grating cheeses (parmesean, pecorino romana).

Also, how you buy your cheese may be significant; many packaged pre-shredded cheeses contain cornstarch to reduce clumping, which might help to reduce the 'greasiness' of the final item (although adding carbs).