Mix paprika, brown sugar, chili powder, cumin, salt, black pepper, oregano, sugar, white pepper, and cayenne together in a small bowl.
Massage spice rub into the pork roast. Wrap tightly in a double layer of plastic wrap; refrigerate for at least 3 hours.
Unwrap the roast and place in the bottom of a lined slow cooker. Add water and liquid smoke. Cover and cook on Low until pork is fork-tender, 8 to 10 hours.
Transfer roast to a cutting board; discard liquid in the cooker. "Pull" by tearing meat into thin shreds with 2 forks or your fingers. Discard fat.
Place shredded meat back in the slow cooker; toss with 1 cup barbecue sauce, and heat on Low until hot, 30 to 60 minutes. Serve with remaining barbecue sauce.