Hire a good contractor!
So you want to dig down lowering the crawlspace an then store stuff an kids in it- would be swell if it were not already damp. Some places are better left not to store or play in...

Reducing the dampness, is there plastic on the soil? If there is is the plastic sealed to the foundation walls? If not get 6 to 10 mill and seal it to the walls this stops more moisture from coming up into your home. Consider insulating the floor joist bays an build the kids a play storage house with the kids in your back yard.