Handling raw poultry turns my cuticles black! Whenever I handle raw poultry (chicken, turkey, duck, etc.) my nails, cuticles, and the skin under and around my nails turns black. This happens even after I thoroughly wash my hands, and when working with thoroughly cleaned utensils.

This has been happening to me for over ten years. So far, I've not been able to get an answer as to why, or how to prevent it.
As a student of medicine I've never heard of any condition similar to the one you describe, but then there are still a lot of diseases, conditions and genetic prepositions unknown to man (and to medicine students).
First I thought, it might be an allergic reaction to poultry proteins, but our skin normally doesn't have many immune cells on the outside (but can still result in a rash or excema) and you would probably have a really strong alergic reaction when eating poultry (that does not seem to be the case). 
What I would like to know is how long does your skin keep being black? Does your hands' colour change slowly or in a few seconds to a black? Do you have any other symptoms during contact to poultry or in general?

Anyhow there are few possibilities. Either something's really wrong with the meat you're buying (and I doubt that) or something's not alright with your skin. Maybe you have some kind of weird bacteria growing on your skin. There a lots of different bacteria physiologically growing on everyones skin ... stil this not seem normal to me. 
You should consider seeking professional help and talk to a general practitioner. 
Meanwhile I suggest using latex or nitril glove (you should be able to buy them in a drug store). This should prevent your hands from turning black and they still allow you to handle the meat precisely.