If you have proof of your residency in New York you should be able to acquire New York insurance on your vehicle so that you can register the car you are buying in NYS. Your insurance provider will need to given information concerning your New York driver's license though when you turn in your Florida license for a New York State license.
If possible it would likely be easier to get your insurance and registration set up if you already have your New York license. It is New York State law that resident of NYS must apply for a NY driver's license within 30 days of becoming a resident anyway so if you do this before you purchase the vehicle it could make things easier.
In order to register a vehicle in New York you must have NYS insurance on it. Out of state insurance coverage is not acceptable or valid to register a vehicle in New York.
Insurance coverage in NYS must be a minimum of $25,000/50,000 for injury, $50,000/100,000 for death, and $10,000 for property damage caused by any one accident. New York State is a no-fault state. The liability coverage must remain in effect while the registration is valid, even if the vehicle is not used (except motorcycles).