I think it is really unlikely that the symptoms that you are experiencing are from frostbite. Frostbite, as you know, is cold damage to the skin and underlying tissues. Usually there are white or yellow patches of skin with numbness
of the affected areas. If the frostbite is severe, then there may be black or blue areas as well.
On the contrary, it sounds like what you are experience is just chapped skin. This is a common problem of the skin that is caused by friction or irritations together with dryness. It can occur, as in your cause, when exposed to cold wind or other irritants.
The treatment for chapped skin is very straight forwards, as all you need to do is moisturize the skin well. I suggest a thick, greasy moisturizing ointment, as opposed to a standard lotion or cream, as these have more staying power and will protect the skin for longer. Apply the moisturizer multiple times per day. Also, you should avoid hot showers or harsh soaps as these will worsen the dryness of the skin and prolong the symptoms.
If you believe you do have symptoms of frostbite, please see your primary care doctor
or emergency physician right away.