It may take months or even up to a year to recover from this. The good news is that you will improve over time. It takes patience and you just have to wait. There is nothing you can do to make it disappear faster, unfortunately but know that overtime things should go back to normal.
Botox is temporary and your results will gradually go away in a few months. I'm sorry to hear that you've had disappointing results, but your mouth will return to its original shape.
The good news is that Botox always wears off and muscle function returns to normal if it was a result of the Botox. Usually the unwanted effects do not last the entire time of the normal treatments. Looking in the mirror and trying to force the normal movements sometimes help things improve a bit. What is important is that you and your injector realize how much was used and exactly where it was placed, so you can avoid a repeat problem.
Currently, for the area around the mouth there is not much you can do to speed up the process. Like other Plastic Surgeons mentioned, it just needs time to wear off.
Luckily whether injected correctly or incorrectly the effect of Botox will last 3-4 months, and your facial muscles will return to their original function.
It appears your injector may have injected too medially and superficially - unintentionally treating one of your lip depressors (DAO). The correct depth of injection for the masseter is too deep to affect the DAO. The other side can also be treated for symmetry, or you can wait for the muscle activity on the treated side to return.
Botox is completely temporary, which is a mixed blessing. Any muscle weakening from botulinum toxin will reverse over months. Since the masseter muscle is not a muscle of facial expression (it is a muscle of chewing), there should be no effect on your smile appearance from masseter muscle injection. The masseter muscle is close to the angle of the jaw, and injecting it with botulinum toxin can create a more "heart shaped" appearance to the face. Did you have the DAO injected? That muscle is between the chin and the mouth corner, and injection with botulinum toxin improves a downturned smile. If this injection point is too close to the chin, there can be inadvertent effect on the depresser labii. When this muscle is weakened, it prevents your mouth corner from going down with a smile. The treatment for the uneven smile is either to wait it out, or make things even by treating the depresser labii on the other side.
It sounds like the injector hit the DAO on the right side to turn up the corner of the mouth on that side. Perhaps you only needed it on the one side. This is routinely done in my office but almost always done on both sides. You are right that it affects the way you make a sad face (but why do you want to do that?). You can get the other side done for symmetry or just let it gradually wear off. Steven Weiner, MD, Facial Plastic Surgeon, Destin, Florida