Don't pay too much heed to official deadlines. Most medical schools are rolling admissions and many times, a competitive applicant who doesn't submit an early application isn't accepted to any school.
You don't have any chance with MD schools this cycle so don't bother submitting any MD secondaries. You do have a shot with DO schools this cycle even though you will be a late applicant.
This is my suggestion. Study like a fiend for your upcoming MCAT test on Oct 21st. Take plenty of practice tests and go over all the questions you did, right and wrong. Aim for a score above 28. As soon as you take your MCAT on Oct 21st, start pre-writing DO secondaries. You can find the secondary prompts in the school specific threads for this cycle on SDN. If you can submit your secondaries before your MCAT scores come out then do so. Otherwise submit your secondaries right after your MCAT scores come out in late November.
Since you are in New York, I recommend the following schools (assuming you will be applying with a good MCAT, good GPA, good ECs, DO shadowing experience etc.):
1. NYIT-COM
2. Touro- NY Harlem
3. Touro- NY Middletown
4. Rowan
5. UNECOM (school with heavy regional preference so contact them and see if your location qualifies as part of their region)
6. Liberty-COM
7. VCOM (all three campuses)
8. William Carey-COM
The 10 schools on this list should be enough but feel free to add more or substitute some out.
