I was in an accident and have pain in both my upper and lower back. I also have stinging pains in my feet that I never had before except if my foot had fallen asleep. Is that related to the back pain, even though I don't have numbness in my legs?

Possibly. Which part of the foot? How often? Consistent in location suggests nerve involvement from lower lumbars.
Maybe. There could be an injury to your vertebrae or disk bulge. This would require an exam and possibly an MRI or other tests to make a true connection between the pain and your symptoms.
I believe. It is back related. Speedy recovery.