Toss the two cabbages, the carrot and all of the bell peppers together in a large mixing bowl; set aside.
In a small bowl whisk together the vinegar, oil, sugar, celery seed, mustard and tarragon. Taste and season well with salt and pepper.
Drizzle the dressing over the veggies, tossing as you go. Coat evenly but do not over-dress. You might not use all of the dressing. Your Carolina Coleslaw is ready to serve but improves upon sitting for about 30 minutes to an hour. Slaw can also be refrigerated in an airtight container overnight. Serve at a cool temperature but not cold.