Replaced tile grout-soaking up moisture and comes out on my hand
The grout has not cured.  A couple of possible reasons are that perhaps it was old and deteriorated or it was improperly mixed.
Either way, at this point it's unlikely to cure properly.  I'd pull it out and re-do the grout making sure to use fresh material that is mixed properly.  Always follow the manufacturer's instructions for curing time and conditions.