Yes it is possible to be "allergic" to cold although it is more often refered to as hypersensitivity to cold.

Your post sounds like it could have been written by me. I have the exact same problem although in my case Claritin does not work. I also get better in 24 hours when I get a bad case of it and to make the coincidence even more arre, were the same size too. LOL

I react to sudden temperature changes, air-conditioning, walking from outdoor to indoor in winter and my nose runs when I eat hot food at work or just because I'm cold. I can be miserable all day at work in an air-conditioned office but perfectly fine when I get home where there is no A/C.

I did a some research on the internet and found out about a condition called non-allergic rhinitis or Vasomotor rhinitis. It causes a runny nose, sneezing, and nasal congestion for the reasons I listed above and more. I also found information about hypersensitivity to cold but most people seem to get hives rather than runny nose.