Seasoning and baking the drumsticks halfway before brushing on the BBQ sauce gives time for the flavor to permeate and the skin to start to brown, then adding the BBQ sauce in increments gives each layer time to bake in.
Set a baking rack on top of a foil-lined baking sheet and spray it with cooking spray. Cut excess skin off the drumsticks, if desired. Mix spices and olive oil in a small bowl, then place drumsticks in a large bowl and toss them with the olive oil & spices to coat. (Use your hands here to ensure that each piece of chicken is coated evenly.)
Arrange the seasoned drumsticks in a single layer on the baking rack, then bake them at 375 degrees for 30 minutes. Use a silicone brush to baste each with BBQ sauce, use tongs to flip, and brush sauce on the other side.
Continue cooking for another 30-45 minutes*, flipping drumsticks and basting on both sides with another coat of BBQ sauce every 10 minutes for the first 30 of those. Serve immediately.
Notes
Some of my drumsticks here were extremely large, so they took the whole 75 minutes to cook through. If you have smaller drumsticks you might just need the hour of cooking time; check for doneness with a meat thermometer.