How can I correct an asymmetrical smile that droops on one-side only? (PHOTOS)

The asymmetry that you have is actually quite common. The right side of your face has stronger muscles than the left and they are pulling more. There is nothing to fix here. There is no good non-surgical or surgical procedure that can truly address the dynamic nature of your concern. Sometimes doing nothing is better than spending a lot of time and money getting procedures that can leave you worse off. Consult with an experienced lip specialist for a complete range of options that are appropriate for your situation.
Gary Linkov, MD
Lip Specialist / RealSelf Lip Lift Medical Reviewer
Manhattan

there is minimal dropping here but there is asymmetry which can be improved most aggressively with a facelift + buccal fat pad removal, or consideration of fixing your bone structure with maxillofacial surgery. non invasively facetite + buccal fat pad can be done plus threads, external heating with morpheus, and fillers to balance and harmonize.  if you are looking for "perfection" you will not find it. but you can get subtle improvements. this is your natural asymmetry that needs regular maintenance treatments.

The best thing you said is that you don't mind your face at rest. That is great in itself. Our skeletal structure and our muscles are different on both sides of the face and body. That's why we have left or right dominance. The face shows exactly the same thing, the muscles are stronger on one side versus the other. No surgery or fillers are needed here. You can actually take the time and exercise the side of the face which you just like in order to bring it to the same condition of the opposite side. I hope this helps and good luck from New York City plastic surgery