TOMS RIVER, N.J. — Diane Burstein spends her days sifting through apartment listings and disaster paperwork and her nights lying awake with worry, her daughter and grandson sleeping feet from her in a cramped hotel room.

Three months after Superstorm Sandy destroyed their home, the Bursteins are among at least 3,500 families displaced in New York and New Jersey who have been living in hotels and motels. Their hotel stays, funded by the Federal Emergency Management Agency, expire every two weeks, leaving them in a constant state of anxiety over whether they’ll be pushed out onto the street.