I would add more material gradually as needed. Since you're spot cleaning every day you really shouldn't have big odor issues unless the area gets waterlogged or something.
I wouldn't bother trying to keep it dry. Store it where convenient, and if that means it's outside and not covered by a roof of some sort, do not tarp it/cover it, as all that does is trap moisture in the pile.Can you store then where they will stay dry?