Pclark, I don't think I've ever heard LES termed that way, i.e. "The farther it gets away from the lake the drier it is".
Ususally LES gets inland about 15-20 miles or so and as the available moisture content is "wrung out", and then it stops, or lessens to a point of not even being worthwhile of a LES event. The available moisture is well, pooped out and then cannot fall anymore, or actually lessened as it reaches a certain point and then dissipates completely.
There's a point inland from the lake where the magic really starts to happen, I'd say about 2 or 3 miles away from shore - at that point the air has been affected by forces of geography, tree cover, changes in temperature between land and sea, etc prevailing wind patterns etc.. and then all Heck breaks loose and the buildup is too strong to stay in the clouds and WALA! You get some serious accumulations in a short period of time!
LES typically is most pronounced in areas 2 miles inland and 15 miles away from the lakeshore, after that point it's all well, "Wrung Out" but I've never noticed an amount of moisture different from shoreline and further from the lake.