Republican primary voters rewarded loyalty to President Donald Trump in primary races Tuesday, ousting South Carolina Rep. Mark Sanford, a vocal critic of the president, and nominating Trump ally Corey Stewart for a U.S. Senate seat in Virginia. Together, the two primary results were a sign of how thoroughly Mr. Trump has come to dominate the Republican Party.

Ms. Arrington will face Democrat Joe Cunningham, a lawyer and yoga-studio owner in Charleston, who carried his infant son with him to vote Tuesday. Mr. Cunningham filmed an ad last year during the total eclipse, saying that Mr. Trump was a darkness falling on democracy. When Mr. Trump tweeted his endorsement of Ms. Arrington over Mr. Sanford on Tuesday, Mr. Cunningham responded, “Shame that neither will be in Congress next year. See you in D.C.” The five-way South Carolina GOP primary for governor is headed to a runoff between incumbent Henry McMaster and 39-year-old entrepreneur John Warren. Mr. McMaster is a fixture in state politics and was among the first elected officials to endorse Mr. Trump in the state’s influential 2016 primary.

Ms. Wexton defeated a field of political novices, contrary to a trend in other states of voters favoring outsider candidates over insiders in Democratic primaries. In a district near Richmond, Democratic voters chose former CIA agent Abigail Spanberger to run against GOP Rep. Dave Brat, the conservative who upended the political world in 2014 by defeating then-House Majority Leader Eric Cantor in a primary. Democrats are convinced Mr. Brat is potentially vulnerable because Mr. Trump won the district, in Richmond’s suburbs, by just six points.

In Nevada’s primary elections Tuesday, battle lines were drawn for the election of a new governor to replace term-limited Republican Gov. Brian Sandoval. The winner of the Democratic primary was Steve Sisolak, a Clark County commissioner backed by former Senate Democratic Leader Harry Reid, a powerful figure in the state. He defeated fellow county commissioner Christina Giunchigliani, who was endorsed by Emily’s List, the national political group that supports women who back abortion rights. Mr. Sisolak will face Republican gubernatorial nominee Adam Laxalt, Nevada’s attorney general. In Maine, Republican voters picked as their gubernatorial nominee Shawn Moody, a businessman and ally of outgoing GOP Gov. Paul LePage. The top vote getters in the Democratic primary were Attorney General Janet Mills and lawyer Adam Cote. But because neither reached 50% of the vote, the final results won’t be known until next week because of a new voting system being used for the first time.

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