If you're into castles, then the Loire Valley in/near Tours, France is a winner. There's a ton of french castles and Chateaux around there to take in. Most (except the big one whose name escapes me) you can do in just a few hours without missing too much, or you can spend all day and explore the grounds. You'll definitely need a car to get around, though. Or take a tour bus, but that forces you into their schedule.
The Bordeaux region is nice. There's quite a few medieval towns in the area, plus its share of chateaux and the ever present vineyards. The weather, even in the summer, isn't too hot. Its not too far (an hour drive?) from the Atlantic coast and resort town of Arcachon (that hosts the largest sand dune in the world) and Lacanau-Ocean (which is one of the premiere wind surfing spots in the world, or so they claimed)
Driving in France is somewhat maddening, once you get off the toll free ways (which were pretty dang pricey, from what I remember). All sorts of round-a-bouts, and the directions only point to the next closest town. (so directions on the provincial roads are somewhat like: Town A to Town B, to Town C, then get off on such and such road.
Other possibilities (still sticking with castles, etc) are the Rhine River in Germany. There's a boat service up and down the rhine that touches a lot of the small towns and goes past a huge number of castles. Or, for a slightly more touristy, there's the Bavarian castle of NeueSchwanstein (and its 'sister' palace, Linderhof) in southern Germany (also close enough to visit Austria, I highly recommend Tyrol area for something cool, off the beaten path, and great if you're into hiking, which I don't think you are)
Rome definitely is not for the walking adverse. Its hot, crowded, cobblestone, and just far enough apart for all of the things you want to see that walking is a stretch, but too close (and parking impossible) to bother with the metro or renting a car (though taxis are abound).
Southern Italy is nice (the Sorrento/Amalfi coast area). You'd have to go see Pompeii, of course, and that isn't stroller friendly. You need a car if you're going further south than Sorrento, as the local train stops there.
You could easily spend a week in Venice, but that is a 'must walk' city (no roads, only canals and walking paths). June probably isn't too hot, but if there's a heat wave, there's little/no air condition and it's humid. It is, by far, one of the most unique cities I've ever been in (quick! go before the oceans rise and its all gone!)
If you just want to 'hang out', try some of the all inclusive resorts in Turkey. I hear the service is fabulous for a reasonable price (and might be on your timeshare list).
The greek islands (also another possiblity for timeshare spots) are fantastically beautiful.
And finally, you should look to see if your timeshare week can get you a discount on a cruise in the mediterranean or something. Scenery is great, you see a lot of neat places, and most of the excursions are bus/walking tours and are made for the stamina of old folks.