How can I fix an overbite and severe spacing between teeth, preferably without traditional metal braces? (Photo)

A DNA Appliance and Epigenetic orthodontics is the ideal solution.  It is removable and only worn 12-14 hours/ day.  It will widen your smile and correct the backward inclination of your front teeth.  It is orthopedic and orthodontic treatment combined.  It will quickly and painlessly improve your smile without braces.

If the bite can be corrected without causing harm to facial aesthetics, then braces are indicated. However, it seems that you might have a skeletal issue (I see lack of upper lip support) and moderate overbite) which can on luck be corrected by braces then jaw surgery.  

It is understandable that you would want to avoid braces if you can but I would strongly suggest you go talk to an orthodontist.  This is by far and away your best option.  You will end up spending a lot of time and money and end up with a poor result if you try other alternatives.  In the long run you will be glad you did it the right way.  Good luck!