Since I do not have an entire picture of your face, it is difficult to an adequate assessment of your face. Sagging of the corners of your mouth usually involves loss of volume from the cheeks as well. I would start by replacing volume in the cheeks, using Restylane lyft which can be used deeper and gives great support. Then after that, I would use Restylane Lyft at the corners of your mouth as well. When evaluating the face, I always like to do a global assessment.
I would recommend a Silhouette Suture lift with 6 threads to suspend and lift the mid and lower face and supplementary injection with Radiesse to enhance the chin and soften the fold that disturbs you.