A contract exists where the parties have come to a meeting of the minds; that is, they both agree on the key issues. Having a writing helps prove that the parties have in fact agreed on the issues.
According to the facts presented in this question, it seems to me you and the contractor have signed a writing wherein the key issues have been specifically enumerated. I would need to see the paper which you both signed, but it sounds as if a valid contract exists. Under contract law, you could have a cause of action for breach of contract. Additionally, you may have a cause of action under the consumer fraud act, since home improvement contracts fall under that statute as well. The consumer fraud act provides for punitive damages in addition to other damages.
As mentioned, a review of your paperwork during a consultation with an attorney in which all facts of your case are discussed, is your best course of action; and one I strongly recommend.
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You very likely have a breach of contract claim here, as well as a violation of the NJ Consumer Fraud Act (potentially several violations). The NJ CFA entitles SUCCESSFUL plaintiffs to 3x their damages in recovery as well as attorney fees. Part and parcel of any NJ CFA case is the required notice to the NJ Attorney General's office, Office of Consumer Affairs.
Without seeing your documentation, it's impossible to determine just what NJ CFA claims you might bring. I suggest you contact an attorney familiar with CFA litigation to provide you with actionable advice.
The foregoing is not legal advice, and nothing in the foregoing shall be deemed to create an attorney client relationship. If you feel you need to speak with an attorney regarding your issue, it is recommended that you contact an attorney with expertise in your area of inquiry. The information related above is purely for informational purposes, and should not be acted upon without speaking with qualified counsel familiar with you specific situation and the laws related thereto.