Is it normal to feel dread before starting a faculty position?
It's perfectly normal to feel the way you do. Part of it is likely fear of the unknown; part is fear of a new job with "adult" responsibilities; part of it might even be fear about whether you can do the job (the infamous Imposter Syndrome). Transitions are always a stressful time, and I don't think we give ourselves enough time to adjust. 

This also reminds me of a survey that discovered that people who just got tenure are the unhappiest of all faculty. The similarity is in the striving for a prize that's difficult to get, and then getting it ! In both cases, there's an element of letdown, and "I did all this work for THIS ?" 

One technique that I've found useful when dealing with these kinds of feelings is to try and focus on the concrete rather than the bigger picture: in your case, maybe focusing on logistics issues related to the transition rather than the bigger picture. 

I've also found as a rule of thumb (both personally and with others) that it can take upto 6 months to feel some level of comfort in a new position. That's a long time.