The two sides of my face are very different - what can I do to look more symmetrical around the nose and mouth? (photos)

Dear nicoledh,
generally speaking, dermal fillers are a great option for face contouring and improvement of asymmetry. If you are considering a surgery, I would suggest you to consult a board-certified plastic surgeon. Only after a thorough examination you will get more information and recommendations.
Daniel Barrett, MD
Certified, American Board of Plastic Surgery
Member, American Society of Plastic Surgery
Member, American Society of Aesthetic Plastic Surgery

you are young and look good. a comprehensive approach with threads, fillers, and morpheus treatments would help tighten and balance.  I would do an at home peel now to even complexion and soften texture and tighten pores/fine lines.
For facial rejuvenation,  I would suggest a combination approach of 
1) skin care with products and PRP/microneedling
2) PDT light therapy few times a month with growth factors
3) laser treatments like clear and brilliant regularly and a few times a year Fraxel/halo Viva, Co2/Erbium
4) Sculptra injections a few times a year for collagen production with fillers like juvederm/restylane/belotero/radiesse/bellafill
5) thermage/ultherapy for yearly skin tightening with or without PDO or cone based instalift threads for lifting
6) botox every 2-3 months to stop movement lines and prevent lines at rest
7) deep microneedling RF like Fractora/morpheus8 for tightening of the skin
8) Vascular laser like vbeam/excel V or BBL forever young treatment every quarter (3-4 months)
Best, Dr. Emer.