You poor darling that sounds awful and truly painful, I would speak to the chemo nurses, maybe they will have a solution.
I didn't have any come off but they were pretty sore especially the middle and index fingernails on both hands, then I got PN and they just felt weird and sort of more painful, so I had to cut them short.
I painted mine with nail hardener, no dark polish as I did not like the look on me. It oddly helped to ease the pain?
Seven months after finishing chemo, I have two fingernails that are ridgy and split a fair way down but are growing out. My other nails look nice.
Good luck, I hope you recover soon.