Nitpick: the official name of the state changed from the “State of Rhode Island and Providence Plantations” to the “State of Rhode Island”.
I work with state and municipal governments all the time. The “State of…” or “Commonwealth of…” or “City of…” is part of the legal name.
Here in Connecticut, there is a “City of Groton” and a “Town of Groton.” (Counterintuitively, the jurisdiction of the city is surrounded by that of the town, because the town is a sub-county division, what some other states call “townships.”) Anyway, the “City of Groton” and “Town of Groton” are two different legal entities.
So if you are talking about a geographical area, you can certainly refer to Connecticut or Rhode Island, but if you are talking about official names, then you have to add the “State of…”, etc.