Make seasoning mix: Preheat air fryer to 400°F for 5 minutes. Meanwhile, stir together sugar, salt, smoked paprika, black pepper, garlic powder, onion powder, dry mustard, and cayenne pepper in a large bowl. Stir in oil, and make a paste.
Toss chicken in seasoning: Add drumsticks, and toss to coat evenly with spice mixture.
Cook chicken in air fryer: Carefully line the air fryer basket with aluminum foil. (The basket will be hot from preheating.) Place drumsticks in an even layer in fryer basket, and cook until chicken skin is rendered and charred in spots and a thermometer inserted in thickest portion registers at least 165°F, about 20 minutes, flipping drumsticks halfway through cook time.
Add barbecue sauce: Brush drumsticks all over with barbecue sauce.
Finish cooking chicken: Continue to cook until drumsticks are glazed, about 3 minutes.