In a blender, combine green onions, garlic, jalapeño, lime juice, oil, sugar, allspice, thyme, cinnamon, salt, and water; blend until smooth. Set aside ¼ cup for drizzling after baking.
Place chicken in a large zip baggie. Pour remaining marinade over chicken and roll it around in the bag to coat. Zip it closed and refrigerate, turning once or twice, at least 2 hours, or up to overnight.
Preheat the oven to 375℉.
Place the drumsticks in a 9x13 inch pan and pour marinade on top. Bake until chicken is cooked through, 50 to 60 minutes.
Heat the reserved marinade and drizzle on top of baked chicken for serving.