How can I explain to my cousin, that I want to try and help take care of  her baby, and have time to dedicate to writing?

Nobody in their right mind would let you anywhere near a baby. You're a degenerate. Get help. 

the same way you explained it to us

Just tell her that you’re happy to help, and tell her what times you’re available. You have no obligation to care for her child, so any help you give her is a favor that she should be grateful for. If you like to write, then set aside some time to focus on that. It’s HER kid. She is the one who must give up her hobbies and free time for the baby, not you. It’s nice that you offered to help her though, just don’t let her become too dependent on you. 

You make up your schedule of  times you are available and tell her what arrangement you'd prefer and have the conversation. The old arrangement is over and a new one must be reached.  

Tell her you're available 2 hours a day, or 5 hours a week, or give her hours, "I'm available between 9am and noon every day" or whatever works with your schedule. You don't need to justify your time to her or explain exactly what you're going to do when you're not helping her, but if she pushes for more time or questions you, just tell her the truth. "I have work to do."