At this point, I wouldn't give up on teaching just yet--- that is, if you still WANT to be a teacher.
Yes, the job market in New York is extrememly competitive. (I am an elementary teacher in NY too... but I'm not in the middle of the city like you are.) There are several colleges that are pouring out qualified candidates.... and the population isn't rising as fast as it used to be.
But you have one really good things going for you that can put you near the top of the list for candidates-- You have three years experience. If your experiences were positive ones, principals will LOVE you. It's a lot easier for a young, but experienced, teacher to fall into a position than it is to get somebody started who is just out of college. Your experience is a major plus.
The hard part is making yourself stand out over the other 3,000 candidates. But, honestly, if you REALLY work at it you can do it. (I've done it.)
1. You said you have over three years' experience, which means you already know people who are teachers. See if you can get any of them who know people in other schools to put in a good word for you. Principals often grant interviews to people that are "recommended" by teachers.
2. If you left on positive terms with your last principal, see if you can get him or her to CALL another school that has an open position. A call from a principal will put you straight at the top of the interview list every time. I know it's a majorly hard thing to do-- go up and ask your former boss for a recommendation, but it will help you. Tactfully go up, thank him/her for all of the great experiences within your school. Let him/her know you're looking for another job. Then give him/her a list 2 or 3 schools that you KNOW are hiring (and provide phone numbers). Ask if he/she would be willing to call the principals and recommend you.
3. Walk into schools, introduce yourself to principals, and tell them you're looking for a job... let them know how much you LOVE their school..... and what an HONOR it would be to work there.... Leave the principal a paper copy of your resume. (And, of course, be sure you've filed you application with human resources before you do this.)
Why does this work? Principals (not human resource offices) usually have direct control over who is hired. When a principal has an opening, HR offices send him/her a pack of 25 or so applications. Out of 3,000--- chances are yours won't be there. BUT if you hand the principal a copy of your resume, he/she can easily set up an interview.
Notice-- all 3 tips above-- involve communicating with principals. That's the key to getting a job. Have people recommend you to principals.... and/or go meet principals on your own.... if you wait around for HR offices to call you, you will never get calls.
There's also a good book on getting teaching jobs... it has some good advice... maybe it can help you.... it's at http://www.iwantateachingjob.com
Good luck to you!