Bring wine, sugar, soy sauce, ginger, and garlic to a boil in a saucepan. Reduce heat, and simmer until mixture has thickened to the consistency of honey, 35 to 40 minutes. Let cool.
Using a sharp knife, make three diagonal slashes through the skin and flesh on both sides of each drumstick (this will help the flavors permeate the meat).
Preheat grill to medium-low. (If you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 7 seconds.) Place drumsticks on grill, and brush with sauce. Close cover, and grill 4 minutes. Flip chicken, and brush with sauce. Continue to cook, flipping and brushing with sauce every 4 to 5 minutes, until cooked through, about 20 minutes total. Season with salt. Serve with coleslaw.