Nice hotel somewhere sunny in Europe with DH and boys, 16 and 13, next July?

[Firesidefox]

We have a big wedding anniversary coming up and want to fly somewhere sunny in Europe at the start of the summer holidays next year for an indulgent and properly relaxing week, or maybe four days if it's expensive.

Criteria:
Really nice comfortable hotel to lie by the pool and read books
(Not hordes of people going mad on all-inclusive)
Beautiful spot
Nice beaches
Some culture locally - we don't just want fly and flop, it'd be nice to have a town to spend a day or two in, having a glass of wine in a sunny square and browsing shops.
Good local food - we are quite foodie and eating really nice local food/drink is important to us!

Budget I'm not sure as it has been so long since we've been abroad I've forgotten how much everything costs. Five grand? Am flexible on this as we have a year to save up.

I'm thinking maybe Italy (no idea where but not Positano as we went there a few years ago - loved it but I'd like to go somewhere new), or Greece as the boys have never been. (But would it be TOO hot in July?)

Sardinia? Corsica?

I'm not keen on Turkey, and we've been to Spain a lot so lovely as it is, probably not there.

Any ideas, particularly of nice hotels, vv welcome!

Hopeful bump

We've just got back from a week at Ikos Aria in Kos. There are a few different Ikos hotels across Greece and Spain (and two new ones opening next year).

The hotel is amazing, the service and surroundings are fantastic and beautiful. The pools are great for relaxing with a book with someone constantly bringing you drinks. There are a lot of kids there, but also adult only areas. There are lots of water sports and other activities that might appeal to your kids. And the food menus have been curated by Michelin starred chefs and I was really impressed with the standard.

The only element is doesn't really cover is the access to locality. My DD and I did spend a day in Kos town exploring which was lovely but was a bus in/bus out experience. They do offer the chance to take a mini out during your stay to explore though (but book when you book as we didn't manage to get one when we were there)

Thank you @MerryBloodyChristmasDay , I was actually thinking about Ikos as I've heard such great things about them. I am really glad to hear what you say, and will google at once as what you write sound PERFECT!

Thank you!