What is the best treatment for my pronounced accordian lines that occur when I smile?

Admittedly, these "accordion" lines that are produced with smiling (sometimes referred to as parallel laugh lines around the mouth) are difficult to treat. However, I have found that the combined use of a series "skin booster" filler injection treatments combined with microdroplets of Botox (or Dysport or Xeomin) to be very helpful for this condition and to yield quite gratifying results. Unfortunately, in my experience, surface treatments, like superficial peels, fraxel lasers, and even fractional medical microneedle therapy, with which I have about fifteen years of experience, have proven of limited value for treating these accordion lines.In my Israel satellite facility, where a great many more regulatory agency approved fillers and volumizers are available, I use Restylane Skin Boosters, a product specifically made for this purpose. The injections are made throughout the accordion area and are injected in a grid pattern throughout the entire region. The skin booster product comes with a special injector mechanism attached to the plunger to ensure that the injections have uniform amounts and concentration. Botox is then similarly injected in this same pattern through the same region. The presence of the skin booster material helps to plump the skin and stimulate new, native collagen formation (neocollagenesis). In my Upper East Side Manhattan practice, I have used varying concentrations of Restylane or Belotero Balance in a similar manner for this purpose with good results.The skin booster treatments are repeated at six week intervals for between two and four treatment sessions. The neuromodulator typically entails only one treatment session. Consultation with a board certified aesthetic physician with experience and expertise in the treatment of this area is important.. 

The lines your are talking about when you smile are due to the texture and overall health of your skin and possible loss of natural fat and volume under the lines.  These lines become more noticeable when you are very lean, sun-damaged, or a smoker.  I would focus on enhancing your skin quality (good skin care or resurfacing) and adding natural volume (such as fat grafting) to the region.

Mid cheek vertical lines on smiling have many options for improvement and few options for complete correction.  For non invasive treatments there is radiofrequency tightening, light peels, and certain lasers like Fraxel.  These soften but will not eliminate the lines.  Microneedling, or the newer Aquagold needles which inject filler and botox cause only brief downtime and also improve the lines.  A variety of techniques using injectable fillers like Belotero and Juvederm, sometimes in conjunction with botox, will also improve them further.  Lastly, combining both energy devices, and the fillers and even the ablative lasers will do all that can currently be done.

These lines are always worse on the side of your face that gets more sun and the side you sleep on. Botox has been tried, but can drop the smile. Fillers like Belotero are worth a try, and Fraxel or fractional CO2 might help. I agree, good skin care is a must. Sculptra, Radiesse,  or Voluma if you are gaunt will help too. Microneedling may also be an option. See a board certified dermatologist or plastic surgeon who can review YOUR best options.

Without posted photos very hard to advise between non surgical fillers like Juvederm or Voluma vs surgery... 

Fillers are most helpful with lines and grooves visible at rest. You will not be able to illuminate all wrinkles, especially when you are contracting your facial muscles during smiling. If you have damaged or atrophic skin, it may benefit from chemical peels or LASER resurfacing. Of course, if you expose your skin to direct sun and smoke, the skin damage will continue to worsen.

From your photos there is not much of a visible problem with lines. However the dynamic production of lines when smiling is a natural phenomenon. It should not be altered as it is part of your facial personality. However with age , smoking, and sun exposure there are natural skin changes that cause fine lines, drying and the deepening of these creases that begin to add an aged look to the face. My suggestion first is to manage skin care with good moisturization and avoidance of sun. Fraxel laser and mild skin tightening may also help. Fillers are problematic as you may look to full in this area especially when you are not smiling and Botox or Dysport may alter the character of you mouth and smile is a very unpleasant manner. Take a conservative approach first and see a surgeon with a good eye.