Can braces help with the appearance of my smile and slight overbite or should I consider surgery? (photo)

By seeing your photo, positive findings are: mal-aligned upper teeth, extruded upper anterior teeth, and gummy smile.
so, the answer to your question is, no surgery is required, orthodontic treatment will correct these issues.
invisalign/aligner will give better results as compared to braces treatment.
alignment of upper teeth with intrusion of only upper anterior teeth (no extrusion of posterior teeth) with improve your smile and correct gummy smile also.