How could I fix my underbite without undergoing jaw surgery? (photos)

Your bite is not a tooth issue.  It is a skeletal jaw issue. If it bothers you, you need to consider jaw surgery. Veneers will not correct your problem.  Consult with an oral surgeon to discuss how the surgery is done. 

Your bite is not a tooth issue, you have a protruded lower arch and a protruded chin.  It is a skeletal jaw issue, your jaw would need to be reset back i.e surgery. 
Veneers or braces will fix the estthics of your case, straighten the teeth and fix the bite and occlusion but your jaw is still protruded. Consult with an oral surgeon to discuss how the surgery is done. Another option is to place Onlays on the posterior teeth, open the bite and improve the relation between upper and lower jaw without surgery. 

yes, u r a border line case. its my advice to visit orthodontist and discuss yr concern. as i can see yr retrocline lower teeth. so IPR in lower arch with classIII elastic will correct yr occlusion.