I'm really sorry that it's come to this, and I know you feel trapped. In many ways, you are trapped. But I think you still have a way out.
Please, try to talk to your cousin. You don't have to tell him about your plans, but just tell him that you haven't been doing well in your house, and that you need to leave. You don't have to ask to stay with him, you could just ask for a ride to one of those shelters, or just for a ride to anywhere.
Your dreams, your creativity, your art, what you can bring into the world, nobody else can make realize those without you. You have entire worlds within you, worlds that nobody else has. You have a duty to them. At my darkest moments the only things that kept me on this earth were the thought of my mother and the fact that I still have comics to write. You, too, still have things to create.
Ultimately, I am just a stranger on the internet. I can't make you do anything, and I hope you find peace, whatever your choice. But I think there is still peace on this world for you, and I think it would be a waste to lose the worlds in your head you were blessed with.