Providence is the capital city of the state, should it not have the most variety? I know of places that fit or nearly fit all the places you mentioned and are great, and who are you talking to that can't even point you towards Kleo's or Yoleni's? Sounds like you have some poor sources.

Not sure I expect Exeter and Burrilville to be foodie havens, of course you're going to have to go to Providence, with some exception in Newport, Westerly, and the in betweens.