Seven states are set to host primary elections Tuesday as the nation comes to terms with last week’s Supreme Court ruling eliminating the constitutional right to an abortion.

This week’s nominating contests could offer the first clues as to whether the political landscape has shifted. Abortion is particularly relevant in Colorado, where GOP voters are deciding whether to nominate a rare abortion-rights-supporting Republican for U.S. Senate.

In Illinois, a Donald Trump-backed congresswoman ignited a political firestorm over the weekend by celebrating the overturning of Roe v. Wade as “a victory for white life,” phrasing that her spokesman later called a “stumble” and was meant to be “right to life.”

The primaries will also offer new insight about the state of the Republican Party, with the central issue in virtually every GOP contest being fealty to President Trump and his baseless conspiracy theories. Those Republicans who have pushed back at all, including a senator in Oklahoma and a congressman in Mississippi, are facing fierce challenges.

Democrats have their own challenges. Illinois voters will decide a rare incumbent-on-incumbent primary for a House seat, while in South Carolina, Democrats are picking which candidate will take on Senator Scott this fall.

In all, primary elections are playing out across Colorado, Illinois, Mississippi, New York, Oklahoma, South Carolina, and Utah on Tuesday. Nebraska is holding a special election.

What to watch:

COLORADO

A GOP businessman who has spoken publicly about his support for abortion rights, Joe O’Dea, is running for the nomination to take on Senator Bennet this fall. Mr. O’Dea’s top rival is a state representative, Ron Hanks, who opposes abortion in all circumstances and attended the January 6 rally that preceded the insurrection at the U.S. Capitol in 2021.

Mr. O’Dea said he backs a ban on late-term abortions and government funding of abortions, but the decision to terminate a pregnancy in the initial months is “between a person and their God.”

While Colorado has trended Democratic over the past decade, Tuesday’s top Republican primary contests will show whether far-right candidates are making progress in their quest to take on uncontested Democrats such as Mr. Bennet, Governor Polis and the secretary of state, Jena Griswold, who has led the national fight against 2020 election deniers.

One of them is Republican Tina Peters, a conspiracy-theorist county elections clerk who’s been indicted for tampering with voting equipment and posting data online. Ms. Peters wants to unseat Ms. Griswold as Colorado’s top elections official despite calls from the state GOP for Ms. Peters to suspend her campaign. She’s running against Republican Pam Anderson, a former head of the state’s clerks association and defender of Colorado’s mail-in elections system.

Colorado’s congressional primaries will measure the staying power of a first-term GOP firebrand, Representative Lauren Boebert, in a sprawling western Colorado district that leans more Republican after redistricting. She’s up against a state senator, Don Coram, a hemp farmer and GOP moderate.

In the Republican race to take on Mr. Polis, a former suburban Denver mayor, Greg Lopez, is facing Heidi Ganahl, the lone statewide-elected Republican as a University of Colorado regent.

ILLINOIS

As in most GOP contests, Mr. Trump is a central issue in Illinois’s Republican primary for governor.

Darren Bailey, a conservative farmer who earned Mr. Trump’s endorsement over the weekend and often reads from the Bible in campaign videos, is part of a six-candidate Republican field. His rivals include Richard Irvin, the first Black mayor of Illinois’s second-largest suburb, who had $50 million in support from billionaire Ken Griffin but was heavily targeted by Democrats who see Mr. Bailey as an easier matchup for Mr. Pritzker.

While Mr. Trump endorsed Mr. Bailey, he also campaigned alongside a first-term representative, Mary Miller, who is challenging a five-term representative, Rodney Davis, in one of the state’s two incumbent-on-incumbent primaries.

At Saturday’s rally, Ms. Miller described the Supreme Court’s reversal of Roe v. Wade as “a victory for white life.” A spokesperson later said she had intended to say the decision was a victory for a “right to life.”

Yet the Illinois congresswoman is no stranger to provocative statements. Soon after joining the House, Ms. Miller quoted Adolf Hitler, saying he was right to say that “whoever has the youth has the future.”

Mr. Davis is a powerful, more moderate lawmaker who is the top Republican on the House Administration Committee, which deals with election legislation and the Capitol complex.

Meanwhile, two Democratic incumbents — Representatives Sean Casten and Marie Newman — are facing off for a Chicago-area seat. Also on the Democratic side, about two dozen candidates are fighting to succeed Representative Bobby Rush, the only lawmaker to ever defeat Barack Obama. They include John Jackson, son of civil rights leader the Reverend Jesse Jackson, and Karin Norington-Reaves, who has Mr. Rush’s endorsement.

NEW YORK

Governor Hochul, who was vaulted into office last fall when Governor Cuomo resigned amid a sexual harassment scandal, is trying to hold on to her job.

Ms. Hochul, a Democrat from western New York, is facing challenges from New York City’s elected public advocate, Jumaane Williams, and Representative Tom Suozzi, a moderate congressman from Long Island.

Tuesday’s elections cover New York’s statewide offices and state assembly races, but primary elections for U.S. House seats and the state senate will be held August 23. Those elections were delayed because of a redistricting lawsuit that led a court to throw out new political maps.

Ms. Hochul, who was Mr. Cuomo’s lieutenant governor for six years, promised to restore New Yorkers’ faith in its government after stepping into the office last summer, but she hit a major stumbling block in April, when her handpicked lieutenant governor was arrested in a federal corruption probe.

Mr. Williams, a progressive running to Ms. Hochul’s left, said the governor is either “consistently shamefully out of the loop, or shamefully enabling through her inaction.” Mr. Suozzi, running to Ms. Hochul’s right, says she’s not being tough enough on crime, suggesting she should have gone further to harden the state’s bail law.

On the Republican side, Representative Lee Zeldin is considered the front-runner in a crowded field that features Andrew Giuliani, the son of New York City’s former mayor; a former Westchester County executive, Rob Astorino; and businessman Harry Wilson. Vice President Pence has endorsed Mr. Zeldin, who also enjoys the backing of the state GOP and Conservative Party, but Mr. Trump has stayed out of the race.

UTAH

The Republican primary for U.S. Senate pits one of Mr. Trump’s closest allies, Senator Lee, against two challengers who have spent months questioning Mr. Lee’s ties to the former president.

A former state lawmaker, Becky Edwards, and political operative, Ally Isom, have attacked Mr. Lee as a divisive politician who cares less about governing than about television appearances and winning Mr. Trump’s favor. Unlike Mr. Lee, neither voted for Mr. Trump in 2020.

Both Republican challengers have highlighted the post-election text messages Mr. Lee sent to Mr. Trump’s chief of staff that show his early involvement in efforts to overturn the election. Ms. Edwards has also stood out by saying she disagreed with the U.S. Supreme Court’s decision to revisit Roe v. Wade.

The Senate primary is testing whether Mr. Trump’s brand of divisive politics and conspiracy theories resonates with members of the Church of Jesus Christ of Latter-day Saints, who make up a majority of Utah’s population.

In November, the winner will take on independent Evan McMullin, a former Republican who won backing from the state Democratic Party in April.

MISSISSIPPI

Congressional primary runoffs are rare in Mississippi, but on Tuesday, two of the state’s Republican incumbents are fighting to keep their jobs in runoffs against challengers from their own party.

Representative Steven Palazzo is seeking a seventh term and was considered vulnerable after being accused in a 2021 congressional ethics report of abusing his office by misspending campaign funds.

Representative Michael Guest is seeking a third term. He voted to create an independent commission to investigate the January 6 Capitol insurrection and was forced into a runoff amid criticism that he was disloyal to Mr. Trump.

Messrs. Palazzo and Guest both failed to cross the 50 percent threshold to win outright in their June 7 primaries. Mr. Palazzo is facing Mike Ezell, the sheriff of a coastal county, while Mr. Guest is going up against Michael Cassidy, a former Navy fighter pilot who has highlighted his allegiance to Mr. Trump.

OKLAHOMA

Republicans are picking two U.S. Senate nominees on Tuesday.

A crowd of high-profile GOP contenders is vying to replace a retiring senator, Jim Inhofe, including Mr. Trump’s former Environmental Protection Agency administrator, Scott Pruitt, who resigned from his Washington post under a cloud of ethics scandals. Other candidates include Representative Markwayne Mullin; T.W. Shannon, the state legislature’s first Black house speaker; and Luke Holland, Mr. Inhofe’s longtime chief of staff.

Senator Lankford is facing a primary test of his own that centers on Mr. Trump.

Mr. Lankford, among the Senate’s most conservative members, has faced backlash from Trump loyalists for not embracing the former president’s lies about election fraud. Mr. Lankford is facing a Tulsa evangelical pastor, Jackson Lahmeyer, a political newcomer endorsed by Michael Flynn, Mr. Trump’s former national security adviser.

NEBRASKA

A judge on Tuesday will sentence a longtime Nebraska congressman, Jeff Fortenberry, on campaign contribution charges on the same day voters will decide who should serve out the Republican’s term. Mr. Fortenberry resigned in March.

Republican Mike Flood will be favored to win the election in the Republican-leaning district over Democrat Patty Pansing Brooks. Both are state legislators.

Regardless of who wins the special election, Mr. Flood and Ms. Pansing Brooks will face off again in the November general election. The eastern Nebraska district includes Lincoln and parts of suburban Omaha as well as rural areas.

By Steve Peoples. Michelle Price, Sara Burnett, Sam Metz, Jim Anderson, Grant Schulte, Sean Murphy, and Jeffrey Collins contributed to this report.