Q:

Can you sue for both unjust enrichment and breach of contract in NY?

Generally speaking, no, you can't; you either have a contract or you don't, and the two theories are therefore contradictory. That said, many practitioners will assert both claims in a complaint so in the event that the court finds that no contract existed, their client can still try to recoup his losses under the quasi-contractual remedy of unjust enrichment.