NEW YORK ­— Just after daybreak, under a pink-hued sky, the first voters began to pick their way through the sand and muck that had been the streets of Bay Head, N.J. Sidestepping the occasional dead fish, they steadily found their way to the firehouse, where a huge generator powered one of the few sources of heat in the tiny seaside borough.

From dawn till past nightfall on Tuesday, displaced residents from dozens of storm-smashed communities up and down the New York and New Jersey coastlines streamed home, gathering with their neighbors for the first time since Hurricane Sandy, with one simple goal in mind.