We were in Anguilla in December and Cap Juluca was only open the last couple of days we were on the island (we stayed for two+ weeks at the Four Seasons) - restaurants on the island are fabulous. We are big Belmond property lovers so we tried to book lunch or dinner to check out the property but the service in getting back to us/curt response was not great. We had friends that then went to CJ in January and they said the renovations to the rooms are fantastic but (similar to our experience) overall service was lacking - maybe it has changed now. I would recommend that you post on the Tripadvisor forum for Anguilla and ask the question as it was SUPER helpful in our trip planning and they will be able to provide more comprehensive/updated information.

Anguilla is nice but we prefer more exclusive/quiet places (we beach travel around the world). So, if you are looking for a true 'retreat' without a ton of people here are a couple of comments: Le Toiny has excellent service but a very small beach (recently cleaned up and installed a couple of years ago - until then, there really was no 'beach') - very quiet though and about a 20 minute drive from other restaurants, etc. (which we prefer but many do not). Morning breakfast is served each morning beside your own private pool and the staff/service is/was terrific.
We loved Canouan (Mandarin Oriental or private villas) - we stayed in the private villas with private pool with butler and expansive space - there are multiple beaches there and you really feel like you have the entire island to yourself. However, there is literally NOONE there....we liked that but again, it has to be to your taste. We also love Turks (we go back once a year) but it is definitely busier....however, you can get great, great deals for November there as it is their quiet season.
We also love Bali but that might be more of a hike.....haha.....
Happy to provide more details on any of the aforementioned or other islands/resorts if desired.