What can be used to correct a cross bite and a 4mm overjet? Braces? Invisalign? (photos)

I'm 22 and never had braces. I have two main problems with my teeth, a ~4mm overjet and a crossbite. What could be done to fix my teeth, or at least reduce the problem? I've been suggested surgery, but I'd rather not go that route. I've also been suggested braces with extractions but I would prefer not to have extract my teeth as well. Could my problems be fixed with braces sans extraction? Another suggestion was Inivisalign. Would that be sufficient to reduce my overjet and realign my crossbite?