Orthoganthic Surgery or Braces for Gummy Smile and Overbite?

Each patient is different, there are many  orthodontists and surgeons who work together. The end result should consider your profile, and a gummy smile can also be corrected by a periodontist after the work is completed.

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

It is hard to say which option would be better for you as a full set of records are needed to make a full diagnosis. Here are general things to think about. If you ask 10 orthodontists how to treat a case, it is possible that you will get 10 different answers and they all may be "right". I would suggest that you feel comfortable with the doctor you are choosing and that you ask him/her the pros and cons of each treatment and its outcome and your expectations (as esthetics and timeframe are important to you and your acting career)
Orthognathic surgery is a serious procedure with risks and healing time that you may want to consider. Orthognathic surgery may yield more extreme skeletal movements than TADs can achieve. Depending on the amount of "gummy" smile you have, and how much intrusion and retraction you need, it may be that the 4 bicupsid extraction and TADs are a good alternative to surgery.
I hope this is helpful!