Esther
I have experienced similar pain and trauma to my hands while working in nicu years ago. It is also much worse in wintertime as the air in this area is usually drier (also my lips suffer then too).
At this point I think you should see a dermatologist and take some days off to heal. When you return to work, then after each handwashing, use a lotion immediately. You may have or have developed a sensitivity to a particular soap or ingredient and may have to use your own mild soap. Have the dermatologist recommend a particular lotion perhaps and carry a bottle on your person so you can use immediately after washing. The sanitizing gels, to me, are less damaging to my hands, even with the alcohol, maybe less friction than traditional handwashing. You may also be allergic to latex in gloves, or powder, if you are using gloves at times. So I would also recommend that, after a period of healing, you use a different glove, have your own box if necessary; you may have to investigate what the least allergenic type is. Also do not use hot water.
Hope you feel better soon. Laurie Wheeler, IBCLC, MN, RN Mississippi, s.e. USA 
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