Anyone have any ideas as to what our problem or how much it could cost
us?

Tile is not watertight. Grouting is not the problem or the solution. It
needs a watertight pan that, in your case, appears to be missing
altogether, or at least incompetently installed. These can be made from
metal, PVC, or asphalt tar. The store and contractor advice you're getting
is foolish if it doesn't even recognize this.