In South Carolina, Mr. Graham’s Democratic challenger, Jaime Harrison, a former chairman of the state Democratic Party who has also worked as a lobbyist in Washington, raised $7.4 million in the first quarter, nearly 30 percent more than the $5.7 million Mr. Graham did.

That $7.4 million made up roughly half of what Mr. Harrison has raised since he entered the race last May seeking to be the first Democrat elected in a statewide race in South Carolina since 2006. Mr. Graham’s campaign has a sizable $12.8 million on hand, compared with Mr. Harrison’s $8 million.

Mr. Graham too, has played an instrumental role in the Trump administration’s judicial confirmations as chairman of the Senate Judiciary Committee. As he has positioned himself more closely to President Trump, he has lost his maverick sheen in the eyes of some Democrats.

Kansas

Barbara Bollier — a Democratic state senator running for the seat that will be vacated by the retiring Republican senator, Pat Roberts — drew more than $2.3 million in the first three months of the year, compared with the roughly $240,000 received by her likely Republican opponent, Kris Kobach, the former Kansas secretary of state and one of the Republican contenders, and the roughly $375,000 received by Roger Marshall, another Republican candidate.

Ms. Bollier, who is hoping to be the first Democrat elected to represent Kansas in the United States Senate in more than eight decades, reported $2.4 million in cash on hand, while Mr. Kobach’s campaign reported having roughly $300,000. Though there was speculation that Secretary of State Mike Pompeo, a former congressman from Kansas, would run for the seat, he told Mr. McConnell early this year that he had decided against it.

Maine

Sara Gideon, the Democratic speaker of the Maine House, raised $7.1 million in the first quarter, roughly three times as much as the incumbent Republican senator, Susan Collins, who has come under particular pressure for her vote to confirm Justice Brett M. Kavanaugh to the Supreme Court in 2018.

As her fight to keep her seat continues, Ms. Collins, who has spent more than 20 years in the Senate, has a million dollars more on hand than her opponent — $5.6 million, compared with Ms. Gideon’s $4.6 million, according to the filings.