On Monday night, several left-leaning congressional candidates joined an emergency organizing call with activists reeling from a draft Supreme Court ruling overturning Roe v. Wade. A somber Senator Elizabeth Warren of Massachusetts, opening the discussion, acknowledged that Democrats held control in Washington but were nonetheless “in an uphill battle for change.”

The moment, she said, demanded leaders “who know how to get in the fight and who know how to win.”

Tensions over how to execute on both of those ambitions — pushing effectively for change, while winning elections — are now animating Democratic primaries from Pennsylvania to Texas to Oregon, as Democrats barrel into an intense new season of intraparty battles.

For the first months of 2022, Republican primaries have dominated the political landscape, emerging as key measures of former President Donald J. Trump’s sway over his party’s base. But the coming weeks will also offer a window into the mood of Democratic voters who are alarmed by threats to abortion rights, frustrated by gridlock in Washington and deeply worried about a challenging midterm campaign environment.

Some contests are shaped by policy debates over issues like climate and crime. House primaries have been deluged with money from a constellation of groups, including those with ties to cryptocurrency, pro-Israel advocacy and an intervening national party, sometimes resulting in backlash. And in races that could be consequential in the general election, national party leaders have openly taken sides, turning some House primaries into proxy battles over the direction of the party.

Tuesday night’s Democratic House primary in the Omaha area attracted less of that national fervor, but it may lay the groundwork for a competitive general election. Representative Don Bacon, a Republican representing a district President Biden won, defeated a vocally left-leaning Democratic contender in 2018 and 2020.

Democrats hope to make inroads there this year despite a brutal national climate, and on Tuesday nominated State Senator Tony Vargas, who has emphasized his governing experience and background as the son of immigrants.

Jane Kleeb, the chairwoman of Nebraska’s Democratic Party, said that recent primary contests had been shaped above all by moderate-versus-progressive divisions. This time around, she said, voters appeared focused much less on ideological labels and much more on policy proposals and electoral viability. It’s a reflection of the urgent concerns held by many Democratic voters around the country who, above all else, worry that their party will lose its congressional majorities in Washington.

“There is a less ideological mood — I think that Democrats, especially in our state, feel like we’re fighting for every office we can get,” she said. “People want to win, but I also think the word ‘progressive’ is not enough. Voters are really wanting to know what the candidate stands for and what they’re going to do when they get into office.”

Beginning next Tuesday, the Democratic primary season accelerates, headlined by the marquee Senate Democratic primary in Pennsylvania. Lt. Gov. John Fetterman has consistently led sparse public polling against Representative Conor Lamb of suburban Pittsburgh and State Representative Malcolm Kenyatta of Philadelphia.

The race, in one of the few states where Democrats have a solid chance of picking up a Senate seat, has focused heavily on what it will take to win the general election. Mr. Fetterman promises to improve Democratic standing in rural Trump territory, while Mr. Lamb, a polished Marine veteran, often cites his record of winning in a challenging House district.

That theme has echoed in a handful of upcoming House primaries, highlighting fierce Democratic disagreements over what the party’s candidates need to do or show to win this November.

In Oregon, Representative Kurt Schrader, the well-funded chair of the centrist Blue Dog Coalition’s political arm who has Mr. Biden’s endorsement, faces a challenge from Jamie McLeod-Skinner, a small-business owner and emergency response coordinator who ran unsuccessfully for Congress in 2018.

This time, Ms. McLeod-Skinner has amassed considerable support from local institutions, as well as from left-leaning groups including the Working Families Party (which convened the Monday meeting that Ms. Warren addressed).

Several county Democratic Party organizations in Oregon, ordinarily expected to back the incumbent or remain neutral, endorsed Ms. McLeod-Skinner and urged the House Democratic campaign arm, which is supporting Mr. Schrader, to stay out of the primary. Johanna Warshaw, a spokeswoman for the Democratic Congressional Campaign Committee, noted that the organization’s “core mission is to re-elect Democratic members.”

Mr. Schrader’s supporters and some national Democrats believe he has a better shot in a fall election that may be robustly competitive. But Ms. McLeod-Skinner’s supporters argue that she can galvanize Democratic voters in a year when Republicans have been widely thought to have the edge on enthusiasm.

Democrats should “want a candidate who Democrats are enthusiastic about,” said Leah Greenberg, the co-founder and co-executive director of the Indivisible Project, a grass-roots group. Citing “local frustration,” she added, “Kurt Schrader is not that candidate.”

In a statement, Mr. Schrader’s spokeswoman, Deb Barnes, said he has a proven ability to “bring everyone together — rural, urban and suburban — to find common ground and deliver wins that make a real difference.”

Electability is playing out in a different way in South Texas, where Jessica Cisneros is challenging Representative Henry Cuellar, the most staunchly anti-abortion Democrat in the House, in a district where conservative Democrats have often thrived.

Ms. Cisneros has strong support from national left-leaning leaders, and abortion rights advocates believe that Democratic outrage around that issue will help her in the May 24 runoff and beyond.

“When we defeat the anti-choice Democrat, that’s going to set the tone for the rest of the midterms,” Ms. Cisneros said in a recent interview.

But other national Democrats plainly see Mr. Cuellar as a stronger fit in a more culturally conservative district that may become a heated general-election battleground.

“We ought not have a litmus test of who and what makes one a Democrat,” said Representative James E. Clyburn, the third-ranking House Democrat, who campaigned with Mr. Cuellar last week.

Still, there are sharp divisions over what it means to be an effective Democrat — a dynamic at the heart of high-profile primary battles in recent years, as left-wing contenders defeated several senior incumbents but also faced setbacks, as in Ohio, where Representative Shontel Brown won a rematch against former State Senator Nina Turner.

Next Tuesday kicks off a fresh series of tests concerning what kinds of candidates can excite — or reassure — Democratic voters at a perilous moment for their party.

“In 2018 and 2020 they were rebelling against an establishment that lost to Trump,” said Sean McElwee, the founding executive director of Data for Progress, a liberal policy and polling organization. “Now they want people who will pass Biden’s agenda and hold swing seats, and progressives need to make the case that they are the best chance to do that.”

In Pennsylvania, a House primary for the seat around Pittsburgh being vacated by Representative Mike Doyle, who is retiring, will vividly test that argument. An attorney and former head of the Pennsylvania Securities Commission, Steve Irwin, has amassed the support of much of the party establishment, while Senator Bernie Sanders and Mayor Ed Gainey of Pittsburgh are expected to campaign this week with State Representative Summer Lee, who joined the Monday call with Ms. Warren. Jerry Dickinson, a law professor at the University of Pittsburgh, is also among those vying for the nomination.

In North Carolina, former State Senator Erica Smith and Durham County Commissioner Nida Allam also participated in the Working Families Party call. Ms. Smith, running in the First District, is vying to succeed Representative G.K. Butterfield, who endorsed State Senator Don Davis. Ms. Allam is facing off against opponents including State Senator Valerie Foushee and Clay Aiken, the former “American Idol” contestant, in the Fourth District. There is also a primary in the state’s newly drawn 13th District, which may be competitive in the general election.

In Kentucky’s primary next Tuesday, State Representative Attica Scott, a vocal leader of the police accountability movement in Louisville, is running to the left of Senate Minority Leader Morgan McGarvey in the race to succeed Representative John Yarmuth.

And in the coming weeks, several incumbent House members will face contested primary elections, while t​​he Los Angeles mayoral primary and the recall vote against San Francisco’s district attorney, both on June 7, will gauge the attitudes of typically liberal Californians on issues of crime and homelessness.

Mr. Sanders, who has endorsed in several upcoming primaries, cast the moment as “a struggle about whether the Democratic Party is a party of working families” or one of “wealthy campaign contributors.”

But he also offered a grave warning for his party that has implications well beyond primary season.

Because Democrats have so far failed to pass major pieces of their agenda, he said, “There is now a great deal of demoralization among working people, whether they’re Black or white or Latino or Native American, whatever. And I fear very much that the voter turnout for Democrats will not be very high.”