I do not know the answer at all,... but it sounds like the sensation part of your brain is sending the wrong signals. It reminds me of when I cooked some Martha Stewart tacos, with chili peppers for somebody. Later, the sensations in my hands were really messed up, I couldn't tell if my shower water was hot or cold, it went back and forth, but it felt cold when my legs said it was hot. I thought something was terribly wrong, till I read about chili peppers. It's almost like that same type of thing is happening in your brain.
I've had migraines,... and so has a friend of mine. I had bit my cheek really badly a few days before,... and mentioned it to my friend. My friend said that the back of the cheek area was part of the same nerve signals to do with Migraines,.. that it could trigger a migraine. This was interesting as I had a mild migraine feeling the day after the cheek incident. Maybe your sinus area, cheek, or teeth are involved with this cold sensation.
Anyway, just some ramblings.
First of all, keep in mind that I am unable to diagnose you because I am unable to examine you, this forum is for educational purposes.
The symptoms and story that you provide is consistent with migraine headaches. However, there is no 'test' for migraines, thus much of the workup is to exclude other things that your symptoms may represents (such as tumor, vascular malformation, etc). For this reason I would suggest an MRI of your brain (with and without contrast) to evaluate for any alternative cause. It is not uncommon for patients with migraine to report odd sensory complaints before, during or even after their headaches. It does not sound like the OTC medications are working for you, and I would recommend stopping them. OTC medication if taken more than 2 times (yes only 2 times) a week can cause rebound headaches (which are very hard to treat). I would suggest that you start a migraine preventative medication, such as elavil, topamax, nadolol, etc. (daily) to stop your headaches form occuring (works in 2-3 weeks). I would also suggest that you see your neurologist about possibly starting a triptan, which can abort your headache (should be started after the MRI is done).
I hope this has been helpful.