Have a hunch... When i was running and training hard i would both clench my teeth together and breathe through my mouth when i was straining. The vibration from running with my teeth clenched gave me headaches and the cold air mademy teeth ache. Why don't you try wearing a light sports mouthguard like an under-armour or cheaper knock off and see how you do. If it persists, see a dentist to get checked out.
See below. This is one of those problems where a visit to your doctor is necessary to figure out what's going on. Only after a thorough evaluation, including examination and possibly labs and other tests, can your doctor correctly diagnose you and treat you effectively.