Maine Democrats nominated Sara Gideon, the speaker of the state's House of Representatives, to face off against moderate Republican Senator Susan Collins, one of the Senate's most at-risk Republicans.
But there was a close, ongoing battle in Texas between Air Force veteran MJ Hegar and Black state senator Royce West over the Democratic nomination to go up against Republican Senator John Cornyn in November.
In Alabama, Tuberville, 65, a former football coach, beat Jeff Sessions, a former US attorney general who was fired by the president. Tuberville had 62.1 percent to Session's 37.9 percent, with 73 percent of precincts reporting, the New York Times said. He will take on Democrat Doug Jones in November.
Sessions had hoped to return to the Senate, where he had been a member for 20 years before joining the Trump administration. But Sessions told reporters he would support Tuberville in November.
While Republican stronghold Alabama went for Trump's choice of candidate, Trump's public approval across the country has dropped as the coronavirus pandemic surged through the United States, killing more than 130,000 people and throwing tens of millions out of work.
That is weighing on his fellow Republicans, dimming the re-election hopes of senators in Colorado, North Carolina and Arizona and leaving even senior Republicans in conservative stakes like Mitch McConnell's Kentucky having to work harder than expected to defend their seats.
Republicans currently control the Senate, 53-47. Democrats would need to pick up four seats in the 100-member chamber for a majority if Trump is re-elected, or three if presumptive Democratic nominee Joe Biden wins the White House, giving the party a tie-breaking Senate vote.
"If the coronavirus continues to get worse and the economy doesn't improve, it's hard to imagine any president ... getting re-elected very easily, and it's hard to imagine that president's party doing well," said Joshua Blank, research director of the Texas Politics Project at the University of Texas at Austin.
Although the primary elections in all three states had been postponed from earlier this year because of the coronavirus, they were held Tuesday even as the number of new cases continued to surge in southern and western states. Texas saw a record 10,745 new cases on Tuesday, and Alabama reported a record daily number of 40 deaths from the coronavirus.