A discrimnation complaint can be filed in any US District Court based upon a violation of federal discrimination laws. Whether you can add state claims for NY and NJ is a complex technical question. I would guess that other technical problems will develop, and you will be at a significant disadvantage if you try to fight this on your own. It would be best if you retained an attorney to handle the case in District Court.
Technically, yes (assuming the court has jurisdiction over the private employer). But whether it is a good idea to do so is a strategic question best explored with an experienced litigator licensed to practice before the relevant courts. Which court it is best to file in is also a strategic question. Don't try to do this pro se.
Not legal advice, just my two cents. I don't practice law in New York or New Jersey or hold licensure in either place. If you need legal advice, and I THINK YOU DO NEED LEGAL ADVICE, please consult a lawyer who holds licensure in those states.
Certainly would would want to consolidate your claims in a single Federal Court action at some point, but there could be tactical reasons why at this stage you may wish to have 2 EEOC offices in 2 separate states involved and separately investigating the occurrences in each state.
I suggest you contact an experienced employment attorney and arrange a free consultation to discuss your options.
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This is what's known as a "Complex Civil Procedure" question.
You definitely want to SIMPLIFY your lawsuit so as not to
confuse and bore the judge(s). What agency did you work for?
State (NY) or Federal? If it was a state agency, you would file
it in state court or federal court. I don't like suing a state agency
in it's own court, I would file it federally, that scares them more!
If you worked for a federal agency, then your answer is federal
court. When you file, just tell the story simple and sweet. Add in
all the applicable laws (NY, NJ and Federal) as needed to tell
your tale and show how you were violated. If needed, the
other side or the judge(s) will explain what laws cover your
action, and there might need to be an amended complaint
later. If this is too complex, hire an attorney. Good luck.
THIS ANSWER IS PURELY FOR ACADEMIC DISCUSSION ONLY AND DOES NOT CONSTITUTE ANY TYPE OF LEGAL ADVICE OR LEGAL REPRESENTATION.