I hope the OP hasn't left the building! Most of us are not saying this is impossible, just trying to point out the difficulties.

With the water caching bit: most people hiking the ADT take a road trip to the Utah and Nevada deserts ahead of their hike to cache water. In the case of "Boston and Cubby," when they found their caches consistently with empty containers, and finally (after testing old and new gallon water bottles) found that the plastic was consistently splitting along the bottom seam, rented a car and redid their caches just before they hiked through. Even if you don't plan to follow the ADT the whole way, do read those journals to get a better idea of what you're up against.

I'd like to suggest hiking west to east. I know this means the best scenery is in the early part, but it puts the prevailing winds at your back and will let you cross the Cascades or Sierra and the Rockies before the snow flies. The same is true of bicycling (which I'd recommend for a cross country trip rather than hiking, simply because you are better able to cover the very long distances between water sources out west).