J., you both must be so miserable! I am sorry but this poor child is suffering from some kind of reaction to the soap that you bathe her with and/or the soap you wash her clothes, towels, sheets, etc. with. She can't help but scratch in misery because it probably is very itchy! You must change what you are using, especially if they are products that contain chemicals, dyes, fragrances, etc. These are very bad for young, tender skin and you can help her.
I have seen great results with some botanically based products that are even doctor recommended. I would love to send you some information and even picture testimonials to show you that your little one can be helped. Don't put her or you through any more stress. This is not a "natural progression of behavior", it's just her trying to help herself from being so itchy. The dry skin is the big red flad here and no matter what you put on her hands, she won't be happy until the real problem is addressed.
Give me your email address and I will send you information on a line of botanically based baby care products that are vegan certified and will help immediately! Put vitamin E oil on for now and DON'T use any lotion with mineral oil it. Good luck to you.
K.