Can a chin implant or sliding genioplasty help shape the lower lips? (photo)

I have a receded chin, and I am planning on getting either a chin implant or sliding genioplasty to move it forward. But what bothers me the most is not my chin, but my lower lip. It doesn't align with my upper lip and it drapes which make it look like my lips don't close all the way. I have been told that a chin implant or s.genioplasty might give my lower lip some more support and help it pull up. What other procedures are there for correction of my asymmetric and saggy lower lips?