In most states, the general rule is that any agreement regarding or pertaining to an interest in real property must be in writing and signed by the parties.
However, courts will often look to the conduct of the parties to see if there is a definitive agreement reached, even if the document in question has not been signed.
The course of conduct of the parties (i.e. had the parties acted as if the lease was in effect even without the signature) could be one issue that a court would look at if required to address this matter.
As such, while the black letter rule of law may require a signature of the landlord, other factors may dictate a different result.