Tweet's is classic. I like it, but I wouldn't call it the best.
I like 22 Bowen
in Newport. There are other excellent spots in Newport and Bristol (admittedly, I know NOTHING of Jamestown). I've found Persimmon
to be quite good, but it's not everyone's favorite. The Boathouse
in Tiverton (not far from either Newport or Bristol) is good too.
*Edit* Just re-read your post. Everything I listed is pricey, so Tweets may be a good bet. Also, the Landing
in Newport is very good for a more moderate price range. I LOVE Jack's
in Warren, RI (next door to Bristol).
If you have time on your way to Boston, check out the Waterfront Grille
in New Bedford. It is, the BEST seafood you'll find in that price range. Competitive prices and EXCELLENT seafood (best Scallops in the region). It's also located on the water adjacent to the historic district so you get a great view. If you want good seafood (not just the fried stuff), check out the Waterfront Grille.