I don't know the law in New York regarding yuor situation, but as a landlord I will share with you what happens to me when someone doesn't have the money to pay the rent and they are not happy for some reason or another. I always keep my places in top repair, but sometimes they wreck the place themselves and don't want to stay there anymore. So they find another place and move their stuff out in the middle of the night and don't leave any forwarding address. Of course they forfeit their deposit, but they don't seem to care.
I suppose I could try to report them to the credit rating agencies, but it's too much trouble and I wouldn't make any money off of it. I could try to track them down and take them to small claims court, but that costs money and even if you win you are the one that has to try to get your money, the court does not get it for you. It is very rare to actually collect from a deadbeat renter, so I usually don't bother, unless the place is damaged so bad (especially if done so maliciously) that it becomes worth it.
So what I would suggest is to NOT go to the post office and sign for the letter, and refuse to sign for anything delivered. Find another place and move out of that moldy crappy broken house to something less expensive that you can afford. I wouldn't pay for the back rents because you had a verbal agreement. He'd have a hard time proving otherwise in court since he did not act in the past. So legally he might have a case, but it wouldn't be a strong one and he would be foolish to pursue it.
Now I would never recommend anyone do that to a landlord that has taken good care of your place and given you ample notice to move out and not tried to charge for previously agreed upon discounts, but your current landlord is acting like a cad. Move out, don't tell anyone where you are moving, and don't leave a forwarding address.