I've been advised that my apartment is in short sale and the landlord has stopped
Paying the mortgage. Due to financial difficulties, I paid half rent last month and the check was cashed. The lease expired in March. Does landlard cashing check constitute new rental agreement?

... Does landlard cashing check constitute new rental agreement?

Nope. What you are thinking about is the legal theory of "accord and satisfaction". It doesn't apply in my opinion. He cashed the check because "one bird in the hand is better than two in the bush".

If your lease is up you likely have become a month to moth renter unless the lease had an automatic renewal clause, did it? It's a moot point since short sales can take up to a year and frequently fall through. If the unit is bought by a new person he would have to honor the rental agreement but I don't think you actually have one at this point