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Hello, I have exactly the same thing, severe pain in both sides of my jaw when exposed to cold wind. I believe it is due to TMJ, which is also linked to grinding of your teeth, clenching teeth together, biting your fingernails, basically anything repetitive that puts pressure on your teeth.  I would recommend going to see a dentist who has knowledge of TMJ, they will be able to tell you how severe it actually is.  In my case I got a tooth guard to wear at night to prevent me grinding my teeth together, which has made my jaw more comfortable and less sensitive to cold.