Remove the drumsticks from the bag, wipe off the excess marinade, and discard the marinade. Lightly brush or spray the drumsticks with oil. Grill over Indirect Medium heat until the juices run clear and the internal temperature reaches 180°F, 1 1/2 to 2 hours, turning about every 30 minutes. During the last 15 to 20 minutes, brush occasionally with the barbecue sauce. Serve warm.