Sounds like you could use some parenting instruction around the care of newborns and the basics of the body's healing capacity.
Whatever you are doing that is causing scratches - whether it's wearing your nails too long or wearing rings with sharp prongs or other feature - stop. Cut your nails, smooth them, and don't wear jewelry.
Learn to use your pediatrician's nurse advice line. This is the office that can support you the best because they know your baby the best. Ask for a referral to an infant first aid class and a parenting support group.
I'm sure you're very tired and nervous with a baby who isn't even a month old, but you should have a 1 month check-up scheduled very soon and you can discuss your concerns there.
But most of us have scratched our babies or trimmed their fingernails too short causing some bleeding, that sort of thing. Kids are extremely resilient. But do make sure you are all checked out on safety precautions in the bigger issues - proper use of a carseat with a tiny baby, proper safety on the changing table and in the crib, and so on.