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.
What if I don't use the vehicle?
As long as your registration is valid, your insurance will remain in effect.