Mid thirties corner of mouth skin fold, what are my options?

There are several options you have to correct this problem.  Sagginess around the corner of the mouth or in the lower face is virtually always related to the cheek (malar) fat pad.  I rarely address the lower face without making some correction to the midface and lifting the cheek area.  In your case because you describe your face as "chubby" you do not want to add volume to your cheek, however there is a way to do a lateral lift to your face with filler that will actually make the appearance of your cheek fullness less.  At your age this is the preferred method of treatment that I use, a nonsurgical face lift using filler.  It takes a skillful injector to understand this technique, especially in a patient who has fullness in the cheeks, so be cautious in selecting your doctor.  Then, after doing the lateral lift if there is still a need for correction in the lower face that can be done with filler locally in that area.  I would also like to mention that Ultherapy is an effective treatment to tighten and lift the lower face and jawline and combining these treatment options can be quite effective and achieve a longer term result. 
Thank you for the question.  Hope that helps.
All the best,
Dr. Flashner

Hi- great question. It is hard to tell based on just one photograph from a side view, but you likely have a deepening of what is known as the "nasolabial fold" which is that line of skin you see going from the corner of the nose down to the corner of the mouth and then down onto your chin. As we age, this fold gets deeper and more prominent. This is seen in many people, regardless of their weight or the "chubbiness" of their face. 
There are multiple options to correct this. Fillers may be used to soften the fold and make the blending of your face more balanced. Or, the fillers may be applied to your cheeks to add volume which will naturally lift the tissues of your face and help erase the fold. However, if you are concerned about the chubby appearance of your face then you may not want the additional volume here. Finally, a facelift procedure can surgically help with erasing these folds and is the most effective option. However, you are quite young to have this procedure done, and it is the most expensive option while requiring significant recovery time. 
You may consider seeing a board certified or board eligible plastic surgeon to perform a full facial analysis and to discuss appropriate treatment options in person. 

Looking at your photo, it is clear that you have beautiful features, with nice mid face volume and full lips. In my opinion thread lifting such as  instalift  or novathreads can be very precise option for you. I am opposed to fillers in the lower face in your case. I would highly recommend you to obtain a consultation with an experienced  aesthetic physician/surgeon.  Skin tightening procedures such as Ultherapy or RF microneedling can define the jawline and tighten the submental area.
Best wishes
Dr. Shanthala