The sputter or hesitation could be a sticking EGR valve,the white smoke isnt a good sign,at the very least you are looking at an EGR cooler and oil cooler replacement.This is a common fault with the 6.0l diesel,the oil cooler gets plugged and restricts coolant flow through the egr cooler,which gets hot and melts the solder out of the core,this allows coolant into the exhaust which is the white smoke you see.
Any competent shop should be able to repair it iwth the reppair manuals that are available today.
Ford Master Tech
30+ years in trade,Ford Master Tech,Journeyman Automotive Tech