Of the places I visited, Edgartown was the 3rd largest town on Martha's Vineyard after Oak Bluffs and Vineyard Haven. It has the advantage of not having the large ferries coming and going all the time, but still has a good selection of restaurants and shops. Trying to park near the water might be annoying, but you can walk the length of the town in minutes.
MV-edgartown
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I enjoyed breakfast at the Edgartown diner, although I thought the coffee was off. Another in our party was pleasantly surprised by a Vegan Menu at The Port Hunter, although it's some kind of secret you have to ask for...? My favorite place was relaxing on top of the gazebo at the Chappy Ferry.