Republican voters in Illinois nominated a conservative hard-liner for governor on Tuesday, lifting State Senator Darren Bailey out from a bruising and costly primary that saw spending from three dueling billionaires — including the current Democratic governor, who spent tens of millions of dollars to meddle in the Republican contest.Mr. Bailey defeated Mayor Richard C. Irvin of Aurora, the moderate Black mayor of the state’s second-biggest city, in a race that captured the ongoing power struggle inside the Republican Party. On one side were the old-guard fiscal conservatives who bankrolled Mr. Irvin. On the other side was an ascendant G.O.P. wing that wants to take a more combative approach to politics inspired by former President Donald J. Trump.Kenneth Griffin, a Chicago-based Republican and hedge-fund founder, plunged $50 million into Mr. Irvin’s campaign in an effort to find a moderate Republican who could compete against Democrats in a blue state. But his preferred candidate came under attack not just from Mr. Bailey and other Republicans, but also from the Democratic Governors Association and Gov. J.B. Pritzker, a fellow billionaire and a Democrat. And Mr. Bailey had his own billionaire: Richard Uihlein, a top financier on the right.Gov. J.B. Pritzker of Illinois at a deli in Chicago on Tuesday.Jamie Kelter Davis for The New York TimesThe Illinois race is the most expensive example yet of a high-risk 2022 Democratic strategy of injecting money into Republican primaries to help more extreme G.O.P. candidates in the hopes that Democrats will face weaker general-election opponents.Democrats welcomed Mr. Bailey to the general election by tagging the opponent they had helped engineer as a “MAGA extremist.”“Bailey is far too conservative for Illinois,” said Noam Lee, the executive director of the Democratic Governors Association.Mr. Bailey called Chicago a “hellhole” during one primary debate, was once removed from a legislative session for refusing to wear a mask and has said he opposes abortion even in cases of rape and incest. Mr. Trump endorsed him over the weekend.Democrats also spent money to shape three Republican primaries in Colorado on Tuesday for Senate, governor and the House — and lost in all three.Worried that an eroding national political climate could endanger Senator Michael Bennet, Democrats spent heavily to intervene in the Republican primary. They helped lift up State Representative Ron Hanks, a far-right Republican who marched at the U.S. Capitol on Jan. 6, 2021. A Guide to New York’s 2022 Primary ElectionsAs prominent Democratic officials seek to defend their records, Republicans see opportunities to make inroads in general election races.Governor’s Race: Gov. Kathy Hochul is trying to fend off energetic challenges from two fellow Democrats, while the four-way G.O.P. contest has been playing in part like a referendum on Donald J. Trump.Where the Candidates Stand: Ahead of the primaries for governor on June 28, our political reporters questioned the seven candidates on crime, taxes, abortion and more.Maloney vs. Nadler: New congressional lines have put the two stalwart Manhattan Democrats — including New York City’s last remaining Jewish congressman — on a collision course in the Aug. 23 primary.15 Democrats, 1 Seat: A newly redrawn House district in New York City may be one of the largest and most freewheeling primaries in the nation.Offensive Remarks: Carl P. Paladino, a Republican running for a House seat in Western New York, recently drew backlash for praising Adolf Hitler in an interview dating back to 2021.But the effort failed as a more moderate businessman, Joe O’Dea, won on Tuesday. His campaign celebrated by handing out faux newspapers to supporters at his victory party with the banner headline “O’DEA DEFEATS SCHUMER.”In his victory speech in Denver, Mr. O’Dea pledged to be “like a Republican Joe Manchin” and lampooned the failed intervention by Democrats as “everything that the American people hate about politics.”“It is pure cynicism and deceit,” he said.Illinois and Colorado were two of seven states holding primaries or runoffs on Tuesday, the first races since the Supreme Court overturned Roe v. Wade last week and thrust abortion back to the center of the American political debate.Governor Kathy Hochul speaks to supporters after winning the Democratic nomination Tuesday night.Desiree Rios/The New York TimesIn New York, Gov. Kathy Hochul won the Democratic nomination for her first full term after succeeding Gov. Andrew M. Cuomo, who resigned under pressure over sexual misconduct. In Oklahoma, voters were sorting through a host of Republicans for a rare open Senate seat. And in Mississippi, one House Republican was defeated in a runoff and another survived a right-wing challenger. There were also contests in Utah and South Carolina, including for Senate.Democrats had also attempted to meddle in the Republican primary for governor of Colorado, where an outside group spent money linking Greg Lopez, a former mayor of Parker, to Mr. Trump in a backhanded attempt to elevate him over Heidi Ganahl, a University of Colorado regent.But Ms. Ganahl prevailed and will face Gov. Jared Polis, a Democrat who became the first openly gay man elected to a governorship in 2018 and is seeking re-election.Democrats had also spent in Colorado’s open Eighth District to aid another far-right candidate. The seat is expected to be competitive in the fall.In another closely watched Colorado race, Tina Peters, a Mesa County clerk who has been charged with seven felonies related to allegations that she tampered with voting machines to try to prove the 2020 presidential election was rigged, lost her bid for the Republican nomination to oversee elections as secretary of state.Ms. Peters has pleaded not guilty, and the indictment made her something of a hero to the election-denial movement spurred by Mr. Trump. But that was not enough for her to defeat Pam Anderson, a former Jefferson County clerk.On Tuesday, one voter, Sienna Wells, a 31-year-old software developer and registered independent who lives in Mesa County, cast her ballot in the Republican primary to oppose Ms. Peters, calling her “delusional.”“She says she wants free and fair elections and stuff like that, but if she gets in, she’ll be the one performing fraud,” Ms. Wells said. “It’s awful.”Tina Peters, the Mesa County clerk who is running for Colorado secretary of state, spoke at an event in Grand Junction in June.Daniel Brenner for The New York TimesIn Illinois, an aggressive remapping by Democrats in the once-a-decade redistricting process created a half-dozen competitive House primaries, including two that pitted incumbents of the same party against each other. The races were the latest battlefields for the two parties’ ideological factions.In the Chicago suburbs, Representative Sean Casten defeated Representative Marie Newman after both Democrats were drawn into the same district. Ms. Newman had defeated a moderate Democratic incumbent to win her seat just two years ago. But she has since come under investigation for promising a job to an opponent to get him to exit her race.The victory for Mr. Casten came two weeks after he suffered a personal tragedy: the death of his 17-year-old daughter.In a sprawling and contorted new district that wraps around Springfield, Ill., two Republican incumbents, Representatives Rodney Davis and Mary Miller, were at odds. The contest has involved more than $11.5 million in outside spending. Mr. Davis is an ally of Republican leaders and has benefited from PAC spending linked to Mr. Griffin, the Republican billionaire, and the crypto industry. Ms. Miller was supported by spending from the Club for Growth, a conservative anti-tax group.Representative Mary Miller greeted supporters at a rally hosted by former President Donald J. Trump in Mendon, Ill.Rachel Mummey for The New York TimesMs. Miller won. Her success was a victory for Mr. Trump, who endorsed her months ago in a contest that was seen as the greatest test of his personal influence on Tuesday. Ms. Miller made headlines at a rally with Mr. Trump last weekend, when she hailed the Supreme Court decision overturning Roe as a “victory for white life.” An aide said she had misread a prepared line about the “right to life.”In Chicago, Representative Danny K. Davis, a veteran 80-year-old Black Democrat, confronted a robust primary challenge from Kina Collins, a 31-year-old gun safety activist, in one of the nation’s most solidly Democratic seats.. Representative Michael Guest at a campaign event in Magee, Miss., in June.Rogelio V. Solis/Associated PressIn Mississippi, Representative Michael Guest held off a primary challenge in the Third District from Michael Cassidy, a Navy veteran.Mr. Guest had drawn attacks as one of the three dozen Republicans who voted to authorize an independent commission to investigate the Jan. 6 attack, even though such a commission was never formed. Instead, a Democrat-led House committee is now investigating.But after Mr. Cassidy narrowly outpaced Mr. Guest in the first round of voting, a super PAC aligned with Representative Kevin McCarthy of California, the Republican minority leader, spent more than $500,000 attacking Mr. Cassidy in the final two weeks before the runoff.In Mississippi’s Fourth District, Representative Steven Palazzo was defeated by Sheriff Mike Ezell of Jackson County on Tuesday. Mr. Palazzo, seeking a seventh term, had earned only 31 percent of the vote in the first round and was seen as vulnerable after a congressional ethics investigation accused him in 2021 of misspending campaign funds and other transgressions.In Oklahoma, the early resignation of Senator James M. Inhofe, a Republican who will retire in January, created a rare open seat in the solidly Republican state and drew an expansive primary field.Representative Markwayne Mullin advanced to the runoff and the second spot was still too close to call late Tuesday. T.W. Shannon, the former speaker of the Oklahoma House, Luke Holland, who served as Mr. Inhofe’s chief of staff, and State Senator Nathan Dahm were competing for the second runoff spot. Scott Pruitt, the former administrator of the Environmental Protection Agency, was on track for a weak fifth-place finish.In Utah, Senator Mike Lee, the Republican incumbent, defeated two primary challengers. In a state that is a conservative stronghold, Democrats decided not to put forward a nominee and instead endorsed Evan McMullin, an independent who made a long-shot bid for president in 2016 by appealing to anti-Trump Republicans.Ryan Biller More