I understand that you feel that you should not have to pay. But simply feeling that you should not have to pay does not necessarily make it so. A contract is a contract. If either party could cancel a contract at any time, there would be no point in signing the contract in the first place. Now certain types of contracts do have a right of rescission period (generally 72 hours = 3 days), but these are typically restricted to door to door sales, etc... Certain states have other restrictions (typically for timeshare presentations or health club memberships). But the default rule is that the contract is binding when signed, and there's no restriction in your state that requires a certain cancellation period within any period of days. In other words, the contract is binding when signed, and legally speaking the company could go after you for the full amount owed. The fact that they're willing to settle for less tells me that they're actually a decent company. 99% of the time the "bad" companies that are set out to just scam and get money will not allow for cancelling for anything less than the full amount owed. That's what they can seek under the law. They don't have to perform to "your standards" but rather merely what they promised to do under the specific terms of the contract, which is often far less than what you'd expect.
In short, and I'm sorry to say this, but unless the contract has a specific right to rescind or cancel within a certain period of time, then there would be no right to rescind, and the company could require that you pay everything that you agreed to under the terms of the contract. Again, the fact that they're willing to accept less tells me that they're actually a decent company more focused on their product than profits.
I know this is probably not what you wanted to hear, but it is the law. I hope that clears things up anyway. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (good or better). Thank you, ***** ***** luck to you!