In late March, Dottie Hickey got a call from Luxor Nursing & Rehabilitation at Mills Pond, the nursing home where her sister lived. The 79-year-old was being moved to make space for incoming hospital patients recovering from coronavirus. Ms. Hickey was told the St. James, N.Y., facility had no choice but to take in these patients under a new state policy.

Ms. Hickey said that after a few days she struggled to reach staff for updates on her sister, and after repeated calls, one employee told her why. The nursing home was…