At advance polls this year, voters would receive a ballot from an official, then fill it out behind a voting screen, using a black felt pen to mark an X in a circle next to the name of the candidate of his or her choice.
Tuesday's contests will determine whether Democratic candidates in the swing districts are moderate or more liberal, as well as whether they are shut out of some races altogether.
The most anticipated races are taking place in California, where the state's chaotic "jungle" primary -a system where the top-two finishers proceed to the general election despite party affiliation- could set the stage where no Democrats or Republicans advance to the main contest in 2018.
Newsom announced his candidacy in February 2015 - more than three years before the primary - and has dominated the race in both the polls and fundraising ever since.
California's results, however, may not be finalized for days in some cases, because the state will count any mail ballots postmarked by Tuesday. But, with good reason, most of the national focus is on California.
Republicans hold five of New Jersey's 12 congressional seats.
The retirements of longtime Republican congressmen Rodney Frelinghuysen and Frank LoBiondo have put two reliably Republican districts in the Democrats' crosshairs.
"Get the vote out in California today for Rep. Kevin McCarthy and all of the great GOP candidates for Congress", the president tweeted. While Republican Diane Harkey held a lead with 25 percent of the total vote, she was trailed by a trio of Democrats who had support in the teens - Mike Levin, Sara Jacobs and Doug Applegate - before the next-closest Republican, who was in the high single digits.
But Democrats are optimistic that Menendez will persevere, thanks in part to Menendez's long history in the state as well as enthusiasm in opposing President TrumpDonald John TrumpDem leader blasts ICE chief for plans to speak to "hate group" Christie: "I have not been asked" to help Trump prepare for Mueller interview Papadopoulos's wife asks Trump to pardon her husband in Mueller probe MORE.
There also were contested Democratic primaries for three of Iowa's four U.S. House seats.
California is of particular importance to the Democrats in their quest to take back the House.
The president "touches my heart", she said. Will it make a difference?
"I don't think the Democrats can take the House without New Jersey", said Brigid Harrison, a political science professor at Montclair State University. "Keep our country out of the hands of High Tax, High Crime Nancy Pelosi".
Voters on Tuesday also went to the polls to choose candidates in primaries in Iowa, Mississippi, Alabama, Montana, South Dakota and New Mexico.
Menendez is seeking his third term and has establishment support, including from U.S. Sen. After Hillary Clinton carried the district in 2016, Democrats are hoping to flip the seat in November. On the Republican side, pharmaceutical executive Bob Hugin has poured $7.5 million into his campaign.
It was a different story in MS, where 66-year-old Gladys Cruz wasn't sure which Republican she would support in the state's Senate primary, but she wants whoever wins to firmly support Trump.