Please help with pitiful rescue baby!

I recently purchased two young chickens to keep my young pullet (around 3/4 mos old) company. (Her eggmate had to go live on a real farm cuz he turned out to be a rooster - oops!) While getting the new babies (around 2 mos old), my girls spotted a tiny chick missing most of her back feathers. The man selling the chickens said we could have her; she was the same age as the others but not growing well. Of course I didn't have the heart to say no - either for the chicks sake or my children's.

Honestly, there were a few with bumblefoot, and I think too many chickens for the space, but I live in a neighborhood and I do not want to take on more than I can handle. (Though I only wanted 2 chickens, possibly three, but I can make 4 work.)

Well, we've had her for about 2 weeks now and she seems to eat well (it took a bit of encouraging, and some warm mushy food at first). She drinks and moves about fine. Aside from when she's stressed, her poop is normal. She does seem dehydrated at times, but no matter how hard I try, her water does often end up spilled. I haven't seen any pasty butt, nose drainage, etc. But her skin is now peeling. It gets thick, yellowy, then peels; if you are familiar with cradle cap, that is what it reminds me of. I tried 'bathing' her - damp cloth rubdown. For the past few days I've been applying aquaphor, either plain or mixed with rosemary oil. (I regularly put rosemary oil mixtures on the chicks - it is great for killing and repelling bugs and actually safe to add to food for them.) The aquaphor seems to help, but it is still happening. It is on her legs, back, and wings / wingpits. Also - she has some newer feather growth, but I've also noticed some of her new feathers seem to fall out before they have really become feathers.

I did originally put her in a 'vest' because she shivered so much, but now she seems to be ok and I was afraid the material might be irritating her skin (it's been off for 4 days or so and there is no change to the dry skin).

I'll try to attach a picture - I would appreciate any advice you might have. I've been keeping her and one of the others I got with her inside, though I try to give them a daily field trip to the back yard.