Can I get veneers or braces with edge to edge bite?

Thank you for your question and positing the pictures.
I think your doing the right thing by seeing out second opinions! You can fix your smile with porcelain veneers if they are placed on both your upper and lower front teeth. This can be accomplished by slightly pushing out your upper teeth and bringing in the lower's With porcelain veneers, you can conservatively align your smile including your bite! The veneers can also be used to whiten your smile and repair some of the chips and damage to your upper front teeth.
Seek out an expert cosmetic dentist in your area who can do a proper smile analysis and diagnostic waxup.

You should not do veneers until you have your teeth straightened or you will absolutely destroy the veneers with the mal-occlusion you present with.I highly recommend a second and third orthodontic consultation as it looks like braces could help you tremendously and probably keep you from ever having to do any veneers at all.If the pictures you posted are current and not from age 14, then I feel it could be possible to treat this with Invisalign to expand and straighten without surgery.