An airfoil for a turbomachine such as a gas turbine engine includes a switchbacked passage for conveying cooling fluid, and flow-distribution passages in a forward wall so as to direct cooling fluid from the internal passage for back side impingement upon a leading edge of the airfoil. An inner cooling wall forms a passage for discharging the spent cooling air from a trailing edge of the airfoil after impingement, and cools one of a pressure side and a suction side of the airfoil by way of conduction. The switchbacked passage may have a serpentine form.