A small amount of filler placed by an experienced injector will provide some support to the lower corner of mouth. Botox or Dysport injection into the DAO muscle may help but I usually recommend starting with the filler first.
Both are good lip fillers as long as the treatment is performed by an experienced injector. I work with the Restylane family of products in my clinic and so I use a lot of Restylane Kysse. It gives a very natural looking result. You will also note improvement in lip texture and color after Kysse treatment. The results last up to a year.