Begin by patting the pork shoulder dry with paper towels. In a small bowl, mix together the sea salt, black pepper, smoked paprika, garlic powder, onion powder, and cumin. Rub this spice mixture all over the pork shoulder, ensuring it is well coated.
Heat the olive oil in a large skillet over medium-high heat. Once hot, sear the pork shoulder on all sides until it develops a nice crust, about 3-4 minutes per side.
Transfer the seared pork shoulder to a slow cooker. Pour the apple cider vinegar and chicken broth over the pork. Cover and cook on low for 8 hours, or until the pork is tender and shreds easily with a fork.
While the pork is cooking, prepare the sugar-free BBQ sauce. In a saucepan over medium heat, combine the sugar-free ketchup, apple cider vinegar, Worcestershire sauce, liquid smoke, powdered erythritol, hot sauce, smoked paprika, garlic powder, and onion powder. Stir well and bring to a simmer. Reduce the heat and let the sauce simmer for 10-15 minutes, stirring occasionally. Season with salt and pepper to taste. Set aside.
Once the pork is cooked, transfer it to a large bowl. Using two forks, shred the pork into bite-sized pieces. Discard any excess fat.
Mix in the sugar-free BBQ sauce with the shredded pork until evenly coated. Serve warm.