Prepare grill to high heat.
Combine first 5 ingredients in a large bowl. Add shrimp; toss to coat. Cover and chill 15 minutes.
Cut swizzle sticks at a sharp angle. Thread 1 shrimp on each skewer.
Combine brown sugar and next 8 ingredients (brown sugar through pepper) in a saucepan; bring to a boil. Reduce heat; simmer 5 minutes or until syrupy.
Place shrimp on grill rack coated with cooking spray. Grill 3 minutes on each side or until done, basting generously with glaze.