I wanted to follow up with the hotels we actually used this summer in case others are looking for recommendations.
In Rhodes Old Town we stayed at the Camelot Hotel. The location is on a quiet medieval side street close to restaurants and the main sites in that area. The building itself is probably 15th century, terrific stonework and mosaic floor. The beds perhaps were also 15th century -- quite hard for my taste. Lovely homemade breakfasts each day. A charming place to stay with the sweetest lady running it all.
Lindos Bay: TUI Blue resort
I couldn't resist the location on the beach! I paid slightly more for an ocean view. The back view is not especially interesting.
This is a resort style hotel with many European tourist groups coming through. There are two swimming pools in addition to a terrific beach - calm waters in the bay, not crowded. Buffet restaurant was lackluster. We ate elsewhere for lunch and dinner.
Lindos town is small, charming, and entirely focused on tourists. A huge number of people crowd in every day on buses from the tour boats docked in Rhodes Town. We accidentally drove into town and got stuck for about an hour on the tiny lane among unmoving traffic and hot-and-miserable walking tourists. The next day we came back in the evening (after the buses were gone) and had a lovely dinner and stroll through the warren of shops.
Recommend: The archaeological site of ancient Rhodes' third city: Kameiras.