Instead of putting the stone on the floor of the oven, try placing it on the rack, directly under the baking sheet. Also, make sure that you preheat the stone along with the oven. So far, I have never broken one.
After some experimenting, I found a good way to store the stone in the oven while traveling. Cut four, 6" pieces from a pool noodle. Slit one side of each piece and slip them over the stone.
2011 Silverado 2500HD Duramax, 4x4, crew cab, long bed
Palomino Puma 253-FBS, 27' 5th wheel
1994 19' Class B on Chevy chassis