I had a very similar situation, but not in debt over 10,000 to my landlord and not quite as dire. I ended up leaving prior to being taken to eviction court and leaving the large city I was in and moving to a cheaper area while I still had some cash flow. It absolutely sucked because I was in such a cool area and I was in a place that had decent rent, it was LOW for the area and I still couldn't afford it. I am not sure if I made a mistake by leaving voluntarily. Probably. I have no evictions on my record but terrible credit now.
So, in your situation, you need to deal with the possibility you may be evicted: over 13,000 you may be able to delay things, but it's going to be hard. Evictions in NYC take an incredibly long time, but they do eventually sometimes happen. Over 10K is around the time landlords start going to court for evictions, usually landlords are patient prior to that. The good news is if you aren't in eviction court yet, it takes some time to process evictions if you fight them. Have you looked at free legal help in NY? It would help to know the borough. NY is so big that even with that it will not identify you at all.
Which borough are you in? What sort of jobs are you looking for? Will you take anything or are you only willing to take certain types of jobs? Are you willing to relocate?
If you want help/suggestions with looking for work, message me. Have you contacted NY organizations that help fight evictions?
I also think right now, you need to start looking at your options:
1) Getting a gym membership is the smartest thing you can do when possibly facing homelessness. You can always shower in a gym, shave or apply makeup, and look presentable.
2) Be prepared to get a second mailbox. When applying for work, you will need to list something, you'll need to have government records sent somewhere if applying for assistance.
3) Downsize what you can. You want to be able to fit the most important stuff into something you carry. That means if you have paperwork, you can photograph it, put it on an SD card, and back it up to Microsoft's OneDrive or something similar. Do you have a vehicle you can live in? I am guessing not, since if you are in the 5 boroughs then there's no need for a car.
4) If you can reach out to any family, if you have any family, reach out to them. If you have churches in your area, reach out to them. This isn't the time for being too proud to ask for help from others.
5) If this isn't a rent stabilized place, and you have family, you could just leave before the eviction is on your record, but it may be worth fighting the eviction.
6) Can you get a roommate?
There are a lot of different job sectors that have needs.
If you don't actually want to commit suicide, it will be really hard to do it. Survival instinct is a real phenomenon, people who genuinely absolutely 100 percent want to die with no reservations at all still sometimes are unable to die because of survival instinct. You should plan like you will be unable to commit suicide and not rely on suicide as a backup plan, because it often doesn't work.
I've helped people with resumes before and job searches previously. I am not a professional at that, but let me know if you want help. I may be useless... but perhaps not. This is something I've wanted to try, so if you want to be a test subject for me on helping people out on this, you'd actually be doing me a favor. Message me if you are considering it. I wouldn't charge anything. I know you don't have it.