does a public defender have to fight for you like a paid attorney

Yes and no. A public defender has to offer you adequate representation. The Preamble to the Rules of Professional Conduct states, "As advocate, a lawyer zealously asserts the client’s position under the rules of the adversary system...." That being said, private attorneys usually have far fewer clients and are typically able to devote more time and effort to one's case than a public defender is able to do.