@Anonymous New York requires a New York employer to report all income to New York State (and New York City, as applicable) on the W2. Because of convenience of the employer (taxation of remote workers), New York companies are not allowed to assume that income earned both inside and from outside of New York state is not taxable to New York. That's left up to the taxpayer to do through Allocation.
You will file a part-year return to New York. Your remote income earned in Florida (when you had no physical connection to working for the company in New York), is not taxable to New York, In the New York return, you will use the allocation screens to designate the portion of income that was earned physically in New York (and, presumably, after moving to New York), and the portion of income that was effectively Florida income. It will be very important to maintain documentation of your move (which it sounds like you have done), because New York could try to contest that your "Florida" income is "New York" income. But it isn't, provided that you were a part-year Florida resident, and prior to moving to New York, you did not provide any services physically within New York State. (New York City would be moot. A NYC nonresident is not taxed in NYC).
Please see the consideration of this by kristinelbly's comments in this thread here. It's a bit of material, but she provides excellent background and ways to prove that your income prior to becoming a New York resident would not be taxable to New York (unless you had a physical connection to that work also).
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