This is not a legal site and so, this is not a legal opinion. The short answer is YES, it is an obligation.
Under contract law, once both parties reach a similar accord and agreement regarding the terms of a contract, a contract comes into force.
The long answer: Verbal contracts are only worth the paper they are written upon. This concept is expressed in a common law known as the Statutes of Frauds:
In the United States, a law setting out such requirements is typically called the Statute of Frauds; the name originates from an English statute that was for "the prevention of frauds." The point of the Statute of Frauds is to prevent false allegations of the existence of contracts that were never made, by requiring formal (i.e. written) evidence of the contract. Contracts that do not meet the requirements of Statute of Frauds legislation are unenforceable, but not void.
An 'oral contract' might be valid, but it's not generally enforceable. It's important to note, however, that while not generally enforceable, if you agree under oath that the contract came into existence, the statute of frauds does not apply, and the contract thus become both valid AND enforceable.
There is a John Wayne sensibility to the concept of the 'handshake' agreement. And it might encumber you with a moral and legal responsibility.
Think about it this way: they might well have acted on your verbal agreement, and now that puts them in a difficult situation.