Every case is different, but your problem seems to be what we call a "skeletal deformity" and not just a dental deformity. Exstractio of teeth and TADs can help by Masking the skeletal deformity by just moving your teeth. Unfortunatley this will not help the "retruded" appearance your lower jaw has compared to your upper jaw. Also you will still have your gummy smile. Some dentists will recommend you have gum surgery to remove gum tissue for this, but this is a very poor decision because you will wind up having unatural appearing large teeth. Without examining you and taking measurements this is just speculative, but ususally accurate, you most likely would benefit from Two jaw orthognathic jaw surgery which would involve advancing your lower jaw and moveing your upper jaw up to get rid of your overbite and gummy smile. You still need orthodontics, but this needs to be planned, because by taking out teeth prior to this may compromise the surgery, also aggressive ortho prior to this to mask the problem will all have to be reversed in order to perform surgery.
Please make sure you seek the adice of a Board Certified Oral and Maxillofaical surgeon prior to deciding on treatment