Whisk ketchup, vinegar, brown sugar, Worcestershire, paprika, garlic powder, salt and pepper in a bowl. Reserve 1/3 cup in a clean covered container in the refrigerator before any sauce touches raw chicken.
Arrange drumsticks in a 6-quart slow cooker and coat with the remaining sauce. Cover and cook on low for 5 to 6 hours until the thickest meat reaches 165°F and is tender around the bone.
Heat the broiler and lightly oil a foil-lined rimmed sheet pan. Transfer drumsticks to the pan with space between them and brush with the reserved clean sauce.
Broil 6 inches from the heat for 3 to 4 minutes, turn carefully and broil for 2 to 3 minutes more until the glaze is tacky and lightly browned. Watch continuously so the sugar does not burn.
Skim the cooking juices, pour into a saucepan and simmer for 8 to 10 minutes until reduced to a thicker sauce. Serve 2 drumsticks per person with the reduced sauce spooned alongside.