Are there any non surgical ways to remedy the asymmetry on my face? Should I be chewing or sleeping on one side more? (Photo)

Both implants and fillers in the cheeks can create good results for patients.  Both can build up the cheeks and create symmetry.  You must balance the pros and cons of each procedure and do what you feel is the best decision for you.
Fillers:
Pros: No downtime Reversible Quick
Cons: Temporary More expensive in long run
Implants Pros: Permanent Great for patients with poor bone structure Less expensive overtime
Negative: Risks associated with surgery Possible infection Downtime
I hope this helps!

There are a variety of techniques that can be used to correct facial asymmetry. A more temporary approach includes dermal fillers (e.g., Sculptra, Voluma, and Radiesse), while fat grafting, use of permanent fillers (Aquamid, Bellafill ) and facial implants would offer a more permanent solution. To address your concerns I would recommend having an in person consult with a board certified plastic surgeon who specializes in facial aesthetics

Symmetry of the face can be improved with well-placed fillers (Voluma, Radiesse, Bellafill) and Sculptra. Fat or implants are alternatives. Cost will vary. I recommend getting a formal evaluation with a facial sculpting expert. Best, Dr. Emer