We were going to stay at The Wauwinet but decided to stay at the White Elephant Village instead (there is a White Elephant Hotel and a Village we stayed at the Village)
I highly recommend staying where we stayed. We had breakfast and spent the morning at The Wauwinet and liked that it was close to the Sankety lighthouse but after that we were ready to get back to the White Elephant. Its extremely beautiful but extremely quiet and secluded.
I preferred the White Elephant bc you could bike all over town and the village. The beaches were great around the White Elephant too. If you have any questions on where to stay etc feel free to ask me.