Tom Kean Jr., the son of a popular two-term New Jersey governor, beat six Republican opponents Tuesday to win the nomination to compete against Representative Tom Malinowski, an embattled Democrat accused of ethical lapses, in what is shaping up to be the state’s most competitive midterm contest.Another son of a storied New Jersey political family, Robert Menendez Jr., easily won a Democratic House primary, making it likely that he and his father and namesake, the state’s senior senator, will serve together in the Capitol.Mr. Malinowski, 56, in a swipe at Mr. Kean’s three unsuccessful campaigns for Congress, said: “I want to do this job. He just wants to have this job.”Mr. Kean, 53, who narrowly lost to Mr. Malinowski in 2020, said his opponent had “squandered” the opportunity to serve New Jersey during two terms in Washington.Mr. Kean’s primary opponents had challenged him from the right as they competed for the support of conservative Republican voters aligned with former President Donald J. Trump. Outside a school near Mr. Kean’s home in Westfield, N.J., where he voted on Tuesday with his wife and daughters, a sign parroting one of Mr. Trump’s favorite labels read, “Warning RINO alert.”But just after midnight, he was more than 22 percentage points ahead of his closest opponent in a race that was seen at least in part as a measure of Mr. Trump’s grip on the Republican Party in a state better known for a moderate brand of Republican politics once epitomized by leaders like Mr. Kean’s father, Gov. Thomas H. Kean.The largely suburban Seventh Congressional District is filled with the type of affluent, well-educated voters who helped Democrats take control of the House in 2018.During the last midterm cycle, Democrats in New Jersey flipped four seats — many of which, including Mr. Malinowski’s, are again seen as potential battlegrounds.As he runs for re-election, Mr. Malinowski is facing allegations that he failed to report stock trades as required. He is also running in a district that gained more Republican-leaning towns when the borders were redrawn to reflect demographic changes in the 2020 census.But late Tuesday, Mr. Malinowski’s campaign got a potential boost from an unlikely camp: Republicans hoping to make a home for centrist voters with the creation of a new party, the Moderate Party. The new party, which will almost certainly face legal challenges, has filed nominating petitions on behalf of Mr. Malinowski.Mr. Menendez won in a largely urban district where registered Democrats outnumber Republicans five to one; winning the primary in the Eighth Congressional District, which includes parts of New Jersey’s two largest cities, is often tantamount to victory in the general election.A lawyer making his first run for office, Mr. Menendez had an array of political and union support early on — as well as crucial backing from his father, also a Democrat and the chairman of the Senate Foreign Relations Committee. Just after midnight, when The Associated Press called the race for Mr. Menendez, he was 70 percentage points ahead of his main challenger, David Ocampo Grajales.If he wins in November, Mr. Menendez, 36, will occupy a seat held for a decade by Representative Albio Sires, a Democrat who announced in December he would not run for re-election.Mr. Ocampo Grajales, the son of immigrants from Colombia, said last week that he had entered the race to give Democrats a viable alternative to a handpicked candidate.“It’s not so much who Menendez is himself, but what he represents: more of the same,” said Mr. Ocampo Grajales, 25.The New Jersey congressional candidate Tom Kean Jr., with his family, after voting at Wilson Elementary School in Westfield, N.J., on Tuesday.Bryan Anselm for The New York TimesA Republican primary in a central New Jersey swing district represented by Andy Kim, a Democrat running for his third term in Congress, was the state’s most colorful contest.Bob Healey Jr., a former singer in a punk band who helps run his family’s yacht-manufacturing company, beat two challengers, Nicholas J. Ferrera and Ian A. Smith, with the backing of the Republican Party.In March, Mr. Smith, the former owner of a gym that repeatedly flouted Covid-19 lockdown rules, was charged with driving under the influence, reviving talk of his past conviction for vehicular homicide. He served time in prison for killing a teenager in 2007 while under the influence of alcohol, and has pleaded not guilty to the new charges.“I may have sung in my past about killing someone,” Mr. Healey said during a debate in May. “Ian actually did kill someone.”By midnight, the winner of the Republican primary to face Representative Josh Gottheimer, a Democrat competing for re-election to a fourth term, had not yet been called.Frank Pallotta, a wealthy former investment banker who lost to Mr. Gottheimer two years ago, was 5 percentage points ahead of Nick De Gregorio, a Marine Corps veteran, with about 85 percent of the vote counted.Mr. Pallotta won the endorsement of former President Donald J. Trump in 2020, but not this time. He also had raised significantly less money than Mr. De Gregorio, who had emphasized his service in combat roles in Iraq and Afghanistan.“We did everything we possibly could, and we left no stones unturned,” Mr. De Gregorio said as he watched the returns come in from the Republican Party headquarters in Bergen County. “We went out there and we listened to what the voters had to say.”Mr. Menendez will be a heavy favorite in November against Marcos Arroyo, a housing inspector from West New York, N.J., who was the lone Republican candidate to enter the race.Mr. Menendez has said that, if elected, he would focus on expanding access to early childhood education and issues that affect the cost of living in New Jersey, where property owners pay some of the nation’s highest taxes.“One of the things we’ve focused on — just because we don’t know the climate that we’re going to be walking into, should we win — is trying to think about what Democratic proposals haven’t been passed yet that we can have bipartisan support for,” Mr. Menendez said in an interview on Saturday.Before being appointed to the Senate in 2006, Senator Menendez held the House seat that his son is now seeking. The borders of the district, formerly the 13th Congressional District, have since been redrawn slightly.The state debuted a new electronic voting system in November, but this was the first time that New Jersey voters were offered a chance to cast ballots in person, on machines, during a primary election.Still, with no statewide races on the ballot, turnout was low.“We were told that we were the fourth and fifth people in our district to be voting today,” Gov. Philip D. Murphy said on Tuesday morning, “and they’ve been open for a while.”Shlomo Schorr contributed reporting. More