DMV is always in contact with insurance companies to electronically monitor your insurance coverage and comply with the New York State requirements.
Can I just notify the United States Postal Service about a change of address or do I have to update this info with the DMV too?
You have to keep your address up to date in the DMV. It is not enough to inform the US Postal Service.
Tell me about the mistake of not bringing the proper documentation to a DMV office.