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. John Cox, a Republican businessman backed by President Trump, captured the other spot.

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.