Can I prove that my agreement was breached even if it wasn't written?

Much as I hate to say it, the answer to this question is that "it depends."

Even worse, it depends on a whole host of factors, such as whether your agreement is of the type that must inherently be written in order to be enforceable under New York law (see the Statute of Frauds), and, considering that your agreement wasn't reduced to writing, whether you can actually prove that both sides in fact reached a "meeting of the minds," and therefore, an agreement capable of enforcement.