Heat oil in a large skillet over medium-high. Add short ribs and sausage in batches; cook, turning occasionally, until browned on all sides. Transfer to a plate. (Do not wipe out skillet.)
Reduce heat to medium; add onion, celery, and garlic to skillet; cook, stirring often, until lightly browned, about 5 minutes. Add tomato paste; cook, stirring constantly, 1 minute. Remove from heat; add red wine, scraping bottom of skillet with a wooden spoon to loosen browned bits. Pour mixture into a 6-quart slow cooker. Stir in tomatoes, rosemary, thyme, oregano, and red pepper. Add short ribs and sausage, submerging in tomato mixture. Cover and cook on low until short ribs are tender, about 7 hours.
Transfer short ribs and sausage links to a cutting board. Skim and discard fat from surface of tomato sauce in slow cooker. Discard rosemary. Let short ribs and sausage links cool 10 minutes.
Meanwhile, prepare pasta according to package directions, omitting salt and fat. Cover to keep warm until ready to serve.
Remove meat from bones of short ribs; shred meat, and return to slow cooker. Discard bones. Cut sausage links crosswise into 1/2-inch-thick slices; return to slow cooker. Stir in vinegar and salt. Serve sauce over pasta.