I do get bits of nail or skin breaking away sometimes. I take the sharpest pair of nail scissors that we own and cut as much as I can off, but sometimes I have to leave a stumpy bit because it's impossible to get at or, if it's skin, too painful to cut. I find that it grows out sooner or later if I just ignore it. Got a small one now on my left thumb, but otherwise my nails look good at the moment -- or maybe it's just the dim lighting :p
I don't know if this makes a difference but I do keep my nails short and not long. If you want long nails, are false ones a possibility?
For frazzled hands, I'd be inclined to recommend an antiseptic cream. Put it this way, the one I used when my right hand got seriously frazzled (itching, rash, scabs, etc.) did the job. Took about a month, but each night I'd rub in a tiny amount, let it sink in, rub in a bit more, etc. But that said, if you've got an ongoing allergic reaction, it might not work.