It depends.
a. If you incorporated in Florida, there is no need to change that (other than contact addresses, so the various states can contact you regarding filing requirements, etc.)
However, you will need to register your business in New York, if you are conducting business in New York. Here is a link to the NYS Information on Conducting Business in NY: Conducting Business in New York.
b. If you are taking a salary out of your S-Corp (that is the correct way to do it!) then yes, you will be considered an employee of NJ and should withhold State NJ state taxes while you are living in NJ.
c. If you are asking how to pay your payroll taxes (to which state) anything withheld or paid by the S Corp for/ from your pay for NJ taxes while you are living in NJ will be paid to New Jersey. Since Florida does not have a personal income tax, no state taxes are withheld on the portion of your wages paid while you were in Florida. Amounts withheld for Federal taxes, social security and Medicare will be submitted to IRS, as required for Florida Corporations.
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