If the Judge won't sign your Order To Show Cause I don't think there is anything you can do.
I am a former federal and State prosecutor and have been handling criminal defense and personal injury cases for over 17 years. The above answer, and any follow up comments or emails, is for informational purposes only and not meant as legal advice.
Once an order to show cause is denied, there is little that can be done to stop an eviction. You should try to find out from the Sheriff's office when you are scheduled for eviction and try to consult with an attorney before that day. Without further details it is difficult to provide you with any further information.
The above constitutes general information only and should not be considered legal advice.
One option is to cure whatever default caused the eviction proceeding.
THE INFORMATION CONTAINED IN THIS POSTING IS FOR GENERAL INFORMATION PURPOSES ONLY AND DOES NOT CONSTITUTE LEGAL ADVICE.
THE FURNISHING OF THIS INFORMATION DOES NOT CREATE AN ATTORNEY CLIENT RELATIONSHIP.
AN ATTORNEY CLIENT RELATIONSHIP REQUIRES THE FURNISHING, REVIEW, AND SIGNING OF A RETAINER AGREEMENT.
When the trial court refuses to exercise discretion to continue a stay of eviction, you have the right to seek a further stay at the Appellate Term of New York State Supreme Court. You may request that a Justice of the State Supreme Court (Appellate Term) provide the requisite stay to you so that you may appeal the district court judge's decision to not provide a further stay as an abuse of discretion. This is a complex procedure and not easy to do yourself. You should start by hiring an attorney and if you cannot do that make a visit to the Appellate Term and explain to a clerk at the courthouse what you want to do. Bring copies of all the papers in the landlord and tenant case.
For any more discussion it is important to know the basis for the eviction case.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.
You don't say if the eviction action was based on non-payment or on some other grounds. If this is a non-payment action and you are able to pay the money, you should seek another order to show cause and BRING THE MONEY TO COURT. Have you sought the help of Nassau-Suffolk Legal Services or any legal services to which you may be entitled through your job? Have you approached public assistance for help?
Good luck. If you find that you are going to evicted, it is generally a very upsetting experience. Find a place for any pets that you may have; get your valuables and breakables out before the eviction and leave them with a relative or friend. Make sure to pack up any medication for yourself and the kids and get all your important papers together (birth certificates, social security cards, kids' medical records, court papers).
The people who perform evictions are not like licensed movers that are (usually) pretty careful of your possessions. Generally, in an eviction, those who are moving your stuff out are just tossing your belongings in boxes, and they may box your canned goods, your dishes, your clothing and your kids' toys all together in one big carton. That's why it's a really good idea to move as much out as you can, before they get there.