TOMS RIVER, N.J. — One storm-battered New Jersey county was delivering ballots to emergency shelters Monday while New York City was lining up shuttle buses to ferry people in hard-hit coastal areas to the polls. With the presidential election looming just a week after Superstorm Sandy’s devastation, authorities were scrambling to make voting as manageable as possible while election watchers warned any shortcuts could compromise the integrity of the balloting.

Election officials in both New Jersey and New York were guardedly optimistic that power would be restored and most polling places would be open in all but the worst-hit areas for Tuesday’s election. New York Gov. Andrew Cuomo signed an executive order Monday allowing residents to cast a so-called “affidavit,” or provisional ballot, at any polling place in the state for president and statewide office holders, an opportunity New Jersey was extending to voters as well.

“Compared to what we have had to deal with in the past week, this will be a walk in the park when it comes to voting,” Cuomo said.

Affidavit or provisional ballots are counted after elected officials confirm a voter’s eligibility.