In New York, General Business Law §771 ("GBL") mandates that all home improvement contracts shall be in writing and contain certain terms of payment, fees for services and materials and start and completion dates, among other terms. GBL §771 is a consumer protection statute to protect the consumer from overreaching of the contractor, such as charging for work that was not agreed upon. A Labor Proposal  signed with a certain price proposal could dictate your rights. Check if it has terms about cancelling.

If none of the terms there help you, then using basic contract law.  You have a valid contract by executing the agreement and placing a down payment on it. Also, check if the contractor lists their licensing information on the proposal or contract. If not, then you should complain to the licensing authorities in New York City. New York has taken a strict approach in this area of the law and New York courts have been adamant in their refusal to permit recovery under a contract by an unlicensed  contractor. Vitanza v. City of New York, supra, at 44, 367 N.Y.S.2d 820; Bronold v. Engler, 194 N.Y. 323, 87 N.E. 427. Discovre if  they are licensed by calling the local licensing department. If their licensing is not in order, you have a quick resolution. Also, the consumer affairs office helps consumer with such complaints. Some times they will prosecute. I think, and I hope for you, that this can be resolved. The problem is contractors often change terms and are not the most professional for whatever reason.

There are ways to void a contract, such as fraud in the inducement. Meaning, if they told you one thing leading you to sign some thing else, you should argue to terminate the agreement and get your money back. You can make a claim in small claims court. It's inexpensive, about $20 for filing, and its a quick resolution. Go to an arbitrator not a judge, and you will be heard immediately on your court date.