Yes, that's right.
Well, on this kind of cases, you will need to file a supporting affirmation sr-60, along with other documents.
Would it be okay if I visit another state where my vehicle is insured, even if it is registered here in New York?
If your vehicle is registered in New York, it MUST have a New York insurance. If you go to texas (where you have a Insurance coverage) you'll have to first register the vehicle there, in order to be able to use their insurance.