NEW YORK—All that was left Wednesday of Glenda Moore's flight from Sandy's rising waters was a blue Ford Explorer, its windows broken to reveal two sand-caked children's car seats.

Hours earlier, at the height of the storm, Ms. Moore's two young sons, Connor, 4 years old, and Brendan, 2, were swept from her arms in the Staten Island borough of New York City after the big vehicle became trapped on a flooded road and she unbuckled them from the seats to head for dry land, she later...