When climbing indoors my right hand fingers tips, get really hard, cracked and start bleeding and it puts a stop to my climbing long before my strength gives up. I don't know if it is being worsened by the chalk (ball) but it seems to me that it is.
I have tried exfoliating and moisturising but it doesn't seem to make masses of difference. My hand normally clears up quite well until the next time I am at the wall and then the same happens again.
Do you know if your chalk using a drying agent? Some chalks (metolious super chalk i think do use them) some people can have a reaction to the drying agent. To be safe use the Moon dust/BD white gold or whatever, anything that says 100% chalk