How to fix lip incompetence? (Photos)

You don't specify the type of jaw surgery, but I bet it was lower jaw one with full and total detachment of the mentalis muscle insertions; this muscle is key to push up the lower lip, speak, etc, it seems it was totally made loose and the surgeon did not re-attach it, thus the incompetence and also the aesthetic issue of the double chin you make when smiling (not seen in pics but I know it is there).

Solution: reinsertion drilling one or two tunnels in the bone and suturing with steel wires the tendon of the muscle back to the jaw, not a difficult procedure for a regular genioplasty surgeon with experience in revisions, get assurances yourself (researching) that the surgeon shares this technical planning and has done this before.

It's hard to know without having more information with dynamic motion. Do you thikn ther was nerve damage or this is just what happened? I don't feel you need a chin aug at this time. I would look into maybe lip filler to build up your lips so it's easier to close your mouth.