I understand about nursing school. You have an assumption that it will make a huge amount of difference in your future life, I hope this is the case. And I agree, it will definitely help you be more employable. And at the end of the course, where you are doing clinicals, it would make NO sense to drop out. These programs are difficult to get into and are quite intense.
However, if you do not pay your part, you need to be prepared for another trip back to court, and you need to be prepared for the court system to react saying "Tough! You agreed to it. Pay your share!" as Ohio has done here. We can't say what they'll do, how this situation will be regarded, but you need an attorney and you need to listen to him/her.
The court will not be the least bit interested in your explanations and your justifications about what a terrible person he is, how scared of him you are, how important it is that you stay in school, how you can't live on minimum wage for the rest of your life, etc. They probably will not be wild about the idea of moving him from private school to public school this year. So you need extra income from somewhere, and soon.
The first reason you need it, in my eyes is that you need a good attorney to guide you from here on out. No one is very sympathetic to this "I'm a single mother and I'm in school to make a better life for myself and my child" when you are agreeing to things and then can't do them, and are trying to make it appear you are a victim. They may very well make you pay your half of the school costs. But if you are going to court again, the good attorney is the first requirement that is going to have a cost involved.