Place pork in a slow cooker and season to taste with garlic powder, poultry seasoning, salt, and pepper. Add quartered onion, apple, and 3 celery stalks. Pour chicken broth into slow cooker.
Cook on Low until very tender, about 8 hours. Transfer pork to a large platter and discard juice and vegetables. Shred pork with a fork and return to the slow cooker.
Prepare barbecue sauce while pork is cooking. Melt butter in a large saucepan or small Dutch oven over medium heat; cook and stir 2 cups celery and 1 1/2 cups onion until onion is translucent, about 5 minutes. Add ketchup, water, vinegar, brown sugar, Worcestershire sauce, mustard, liquid smoke, and 1 tablespoon garlic powder; mix. Reduce heat to low and simmer, stirring occasionally, until sauce is thickened, about 10 minutes. Pour barbecue sauce over shredded pork.
Continue cooking pork on Low heat until flavors combine, about 2 hours.
