Place turkey drumsticks in a large shallow dish or heavy-duty zip-top plastic bag.
Combine butter and next 8 ingredients, stirring well. Pour over drumsticks; cover or seal, and chill 8 hours, turning occasionally.
Drain drumsticks, reserving marinade. Wrap each drumstick and 1/4 cup marinade in heavy-duty aluminum foil.
Grill, covered with grill lid, over medium heat (300° to 350°) about 1 hour, turning turkey after 30 minutes. Remove foil, reserving marinade. Return drumsticks to grill, and continue cooking 20 to 30 minutes, basting frequently with reserved marinade.