To prepare pork, soak wood chunks in water about 16 hours; drain.
Combine turbinado sugar and next 4 ingredients (through ground red pepper); reserve 2 tablespoons sugar mixture. Rub half of the remaining sugar mixture onto pork. Place in a large zip-top plastic bag; seal and refrigerate pork overnight.
Remove pork from refrigerator; let stand at room temperature 30 minutes. Rub remaining half of sugar mixture onto pork.
Combine reserved 2 tablespoons sugar mixture, 1 cup vinegar, 1/4 cup water, 1 teaspoon salt, and oil in a small saucepan; cook over low heat 10 minutes or until sugar dissolves.
Remove grill rack; set aside. Prepare grill for indirect grilling, heating one side to medium-low and leaving one side with no heat. Maintain temperature at 225°. Pierce bottom of a disposable aluminum foil pan several times with the tip of a knife. Place pan on heated side of grill; add half of wood chunks to pan. Place another disposable aluminum foil pan (do not pierce pan) on unheated side of grill. Pour remaining 2 cups water in pan. Coat grill rack with cooking spray; place on grill.
Place pork on grill rack over foil pan on the unheated side. Close lid, and cook for 4 1/2 hours or until a thermometer registers 170°, gently brushing pork with vinegar mixture every hour (avoid brushing off sugar mixture). Add additional wood chunks halfway during cooking time. Discard any remaining vinegar mixture.
Preheat oven to 250°.
Remove pork from grill. Wrap pork in several layers of aluminum foil, and place in a baking pan. Bake at 250° for 2 hours or until a thermometer registers 195°. Remove from oven. Let stand, still wrapped, for 1 hour or until pork easily pulls apart. Unwrap pork; trim and discard fat. Shred pork with 2 forks.
To prepare sauce, combine 1 cup vinegar and remaining ingredients in a small saucepan; bring to a boil. Cook until reduced to 1 1/4 cups (about 5 minutes). Serve sauce warm or at room temperature with pork.