Senior housing offer a month to month lease

[Uline One]

Hello,
I am a Senior waiting to move into Sr. Housing in Ft. Lee, N.J. The apartment is not yet ready. I told my landlord and he told me no. A month to month is available in apartment complex in Ft. Lee where I am now living. The new apartment should be ready by end of July. Now I did not sign any new lease, old one up May 31st. They offered me a 60 day written clause with a 1/2 month penalty only.
I wanted month to month. Then I heard about a statute/law in N.J. that applies to Seniors waiting for affordable housing that the Sr. can give his landlord a 40 day written notice/no penalty. I am trying to find out which law that is and if it indeed applies to me.
I am a 66 year old widow. Does your organization know of that law? I did pay the June rent and new security. I received from my landlord a Quit and Vacate notice dated May 23 ending the 5/31st lease. It says I can continue to pay month to month so I am very confused. I have not signed any lease. I want to do the right thing and have been honest with my present landlord from the start. Can I be held financially liable to old landlord? I also have security deposit with present landlord also. I do not know how that works either.
Does your organization handle something like this? Please advise. Thank you.

[sasolis]

Google "senior giving 40 day notice in n.j." and there are several law sites regarding this