I would argue that Sagres is the gold standard in Fall River and a bargain as well.

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Sagres Restaurant
177 Columbia St, Fall River, MA 02721

While not vintage Rhode Island, New Rivers is by far my favorite restaurant in Providence--its been around for 20 years and run by James Beard winner Bruce Tillinghast. Uses local produce/food and I've had things such as cock's combs there (rooster). Daily/weekly menu changes and its casual. Would need reservations.