Clesta Dickson, 86 years old, never married. The retired teacher lives on her own in a tidy apartment on the second floor of Pleasantview Towers, a subsidized apartment building for older adults and people with disabilities in Vienna, W.Va.

When her own parents became frail, she bought a house for the three of them. They died years ago. She has a brother, but he is 82. Without children of her own, she wonders what will happen to her.

“I...