Place a large colander into the sink. Place the coleslaw mix and red cabbage into the colander. Sprinkle lightly with salt, toss, and allow to stand 15 minutes.
Pour the vinegar into a large bowl. Grate apples into the bowl and toss with vinegar to keep from turning brown. Stir in the grated carrots, mayonnaise, sour cream, and celery seed until evenly coated. Season with black pepper to taste.
Rinse the coleslaw mix and cabbage with very cold water; drain, pressing out as much water as possible. Taste, and if still too salty, repeat the process. Pat dry with paper towels. Stir coleslaw and cabbage into apple mixture, and toss to coat evenly with dressing. Chill until ready to serve; stir before serving.