My baby's father does not want custody

He can. I assume you mean by signing a document to that effect? It wouldn't hurt, but it wouldn't necessarily be iron-clad, should he later change his mind. You should talk to an attorney in your jurisdiction to get such a document drafted, and if the attorney agrees on such a course of action.

Handing over custody does not mean he has no rights, even with full custody he will have rights to visitation and he can be made to pay child support even if he gives you full custody

He can not do it without going though courts,