I have a very sad mouth that when relaxed my lips don’t meet and never have. Options? (photo)

It appears that you need volume. There are many ways to accomplish this goal, from fillers to fat transfer, and it will make a modest improvement. Ultimately we cannot change anatomy with these treatments, but can augment what you have. A small amount in the lips, some at the corner of the mouth, and then enough in the cheeks for a slight lift would look nice. 
Hope this helps.
Best of luck,
DrC

from the photos you have provided I would recommend some Botox to relax the mentalis and depressor anguli oris muscles. This would help but honestly, if you were my patient I would not operate until you were ready for a facelift. The opinion that a facelift doesn't correct a sad mouth doesn't make sense. If a sad mouth is becoause of the downturned mouth and jowling, then a face and neck lift is exaclty the answer. The lips not touching can be from other skeletal deficiency such as small chin or receded lower jaw. But I can not comment on that as I don't have that info. Best of luck Dr M