Can I request to have full custody and only supervised visitation for her father and am I able to choose the person?

I have a six month old baby. My daughters father and I were never married.His name is on her birth certificate. I am the one who takes the most care and responsibility for her. The few times I have left her home with him I can tell that he has not taking proper care of her. For example her diaper is so over filled that her pants are soaked and she has the beginning of a rash or she will start to scream within the first five minutes I get home and he has told me she just ate. He has a learning disability which I feel he does not understand the consequences of his behavior. He has not held a steady job all of his life and is currently not helping with any finacial support. I have a notebook of other times he has not responded to her needs.