A midface lift can, in some people, slightly raise the corners of the mouth, but it wouldn't be my procedure of choice. Other options to raise the corners of the mouth are plumping up the cheeks and/or lips with filler, injecting Botox into the muscles that pull the mouth down (depressor anguli oris), or a corner lip lift.
In the right candidate, the corner lip lift can be a beautful thing with virtually no visible scars. I usually do it in older women with thin, caucasian skin. Thicker skin and men do tend to form more scars. Since you said that you have sunken cheeks, this clues me in to the fact that your droopy mouth is a result of fat loss and deflation of the cheeks.
Therefore, (without the benefit of seeing you) I would think that cheek augmentation with injectable fillers or fat transfer would be the preferred treatment for you.