Echoes of Donald Trump’s worst campaign instincts — on gun control and the news media — are registering down ballot in North Carolina, a crucial toss-up state where Senator Richard Burr, a Republican, is in a tight re-election race.

At a private gathering of Republican volunteers last weekend, the senator said he had stopped at a local gun shop and “nothing made me feel better” than seeing Hillary Clinton’s photo on the cover of a gun-rights magazine. Not realizing he was being recorded, the senator drew laughter from the crowd as he added: “I was a little bit shocked at that — it didn’t have a bullseye on it.”

Mr. Burr had to quickly apologize for an “inappropriate” comment when his offensive remarks turned up in a CNN report. His bullseye talk recalled Mr. Trump’s remark in August that “Second Amendment people” could “maybe” find a way to deal with Hillary Clinton and her gun safety agenda if she reached the White House.

Later in the week, in a move that recalled Mr. Trump’s crusade to demonize and black-ball selected media outlets, Mr. Burr’s campaign vindictively banned the state’s main newspaper, The News & Observer, from receiving advance notice of the senator’s daily campaign schedule. Republican officials accused the newspaper of bias in its coverage of the challenge Mr. Burr is facing from Deborah Ross, a Democrat. They are neck and neck in opinion polls. The paper recently reported on Mr. Burr’s extensive fundraising activities and criticism of his poor attendance at Senate committee meetings.