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Sheena Murphy and her husband have long enjoyed weekends in the Hudson Valley, a few hours north of New York City, staying in AirBnB rentals while visiting friends.

When a baby came in the summer of 2015, Ms. Murphy, 36, an interior designer, and her husband, Paul, 37, a game developer, decided to look for a more permanent second home in the area—a getaway in addition to the Brooklyn home they own.

That year they paid…