How to store fresh peaches?
For ripe peaches the main thing is avoid handling them.  They bruise very easily.  I leave them in the box and just slide the box in a shelf in the fridge.  You are only going to get a few days out of them. 

Never in a drawer as too much handling.  Don't stack.  Dry.  I would never wrap or seal to keep mold down but that part is more opinion than hard evidence.