Mikie Sherrill spoke to supporters at her primary night party in Verona, N.J., on Tuesday.

VERONA, N.J. — A former Navy pilot, a veteran White House national security adviser and a conservative pro-gun state senator won Democratic nominations on Tuesday to represent battleground congressional districts in New Jersey, officially earning their spot on the front lines in a state that could determine control of the House.

The day’s political focus remained largely on California and its high-profile statewide races and crowded congressional primaries. Gavin Newsom, the Democratic lieutenant governor, won one of two spots in the primary Tuesday, The Associated Press reported, in his bid to succeed Gov. Jerry Brown. The second spot was still undetermined late Tuesday.

Senator Dianne Feinstein easily won a spot on the November ballot in her re-election bid in what by every indication looks like an easy race this fall — no matter who ends up running against her.

New Jersey could prove as crucial as California in the contest for the House in November. Democratic leaders are hoping to flip as many as four of the five Republican-held districts in the state.

In perhaps the most closely watched contest in New Jersey, Mikie Sherrill, a former Navy pilot and federal prosecutor, won the Democratic nomination in the 11th District, The A.P. reported, where Representative Rodney Frelinghuysen, a Republican, is retiring from a seat that a Democrat has not held since the 1980s. Ms. Sherrill became the latest woman to prevail in a nationwide wave of female candidacies and will now continue a nearly 18-month effort to replace Mr. Frelinghuysen with a Democrat in a district where President Trump’s victory has spawned intense activism.

Bob Hugin, a Republican candidate for Senate, voted in Summit, N.J., on Tuesday.

“We made New Jersey’s 11th district the center of a national movement for new leadership,’’ she told supporters at a victory rally. “Change is led by the people of New Jersey.’’

Ms. Sherrill, who raised nearly $2.4 million even before the primary, will face Jay Webber, a state assemblyman, in the general election.

In the contest for the Senate in New Jersey, the Democratic incumbent, Robert Menendez, won his primary and will face Bob Hugin, a pharmaceutical executive who easily won the Republican nomination.

New Jersey was one of seven states besides California where voters were choosing candidates on the busiest Primary Day of the year. In Alabama, a Republican congresswoman who opposed Mr. Trump’s presidential campaign in 2016, Representative Martha Roby, was forced into a runoff — laying bare the price of dissent in the conservative South even for an incumbent who has worked with the White House since Mr. Trump’s election.

But elsewhere, women candidates had another big night — including a Democratic nomination for governor in New Mexico, a Republican nomination for governor in South Dakota and a win in a competitive Democratic primary for a battleground House seat in Iowa.

In New Jersey, the House races are so important that the Democratic Congressional Campaign Committee made endorsements in four of the primaries to help clear the field for candidates they believed had broad enough appeal to win over the state’s many unaffiliated voters in the fall.

In the southern region of the state, where Representative Frank LoBiondo is leaving his seat in the Second Congressional District, Jeff Van Drew, a state senator, won the Democratic primary. Mr. Van Drew, a conservative Democrat with a pro-gun record who voted against same-sex marriage in 2012, nonetheless won the backing of powerful South Jersey Democrats early on and was able to fend off a challenge…