A series of Election Day contests may serve as a barometer of how far left Democratic voters in New York State want their party to go.Last November, the often-fractious Democrats of New York papered over their sharp differences to celebrate Donald Trump’s defeat, a development that briefly united the party’s relatively moderate leader, Gov. Andrew M. Cuomo, with the state’s ascendant left wing.One year later, New York Democrats are in a vastly different place. Mr. Cuomo has resigned in disgrace and faces the prospect of a criminal trial. President Biden is in the White House, and the center-left politics that propelled his campaign have been embraced by the new governor, Kathy Hochul, and the likely next mayor of New York City, Eric Adams.And all across the state, a series of Election Day contests are setting up fresh tests and tensions over the direction and identity of the Democratic Party.In New York City, Mr. Adams, who is heavily favored to win Tuesday’s election, has already declared himself the face of the Democratic Party, and many national Democrats have elevated him.Mr. Adams, a former police captain who fought for reforms from within the system, has described himself as both a “pragmatic moderate” and “the original progressive.” But he is also a sharp critic of the “defund the police” movement; he makes explicit overtures to the big-business community; and he defeated several more liberal rivals in the primary.A very different face of the Democratic Party may be emerging in Buffalo: India B. Walton, a democratic socialist, who defeated the incumbent Democratic mayor, Byron W. Brown, in the June primary. Mr. Brown, a former state Democratic Party chairman, is now running as a write-in candidate in a closely watched rematch that has become a proxy battle between left-wing leaders and more moderate Democrats.Then there are the Democrats, from Long Island district attorney candidates to the occasional New York City Council hopeful, who face serious opponents in races that will offer early tests of Republican Party energy in the Biden era.After an extraordinary summer of political upheaval, power dynamics are now being renegotiated at every level of government, shaped by matters of race, age, ideology and region. The influx of new leadership has implications for issues of public safety and public health, for debates over education and economic development — and for national questions surrounding the direction of the party.“There’s a battle of narratives in New York,” said State Senator Jabari Brisport, a Brooklyn socialist. “You do have Eric Adams getting elected in New York City, then you have a socialist like India Walton getting elected in Buffalo, right in Gov. Hochul’s backyard. New York is in the midst of finding itself.”The mayoral race in Buffalo between India Walton, center, and the incumbent, Byron Brown, has become a proxy battle between left-wing leaders and more moderate Democrats.Libby March for The New York TimesThe most consequential New York election this year is the race for mayor of the nation’s largest city, which will be decided on Tuesday as Mr. Adams competes against Curtis Sliwa, the Republican founder of the Guardian Angels.Backlash to New York City’s vaccine mandates in more conservative corners of the city, and the prospect of a relatively low-turnout election, inject a measure of unpredictability into the final hours of the race and could affect the result margin, some Democrats warn — but in a city where Republicans are vastly outnumbered, Mr. Sliwa is considered a long shot.The more revealing contest regarding the direction of the Democratic Party is taking place about 300 miles away in Buffalo.That mayoral race is unfolding in raw and divisive terms: Ms. Walton has referred to Mr. Brown as a “Trump puppet” who has become complacent about Buffalo, while his campaign questions her character and paints her sweeping proposals as “too risky” for the city, a message she has cast as fearmongering.In a sign of just how high tensions are running, Jay Jacobs, the state party chairman, sparked outrage when he used a hypothetical candidacy of the former Ku Klux Klan leader David Duke to argue that the party was not obligated to support every nominee, including Ms. Walton. He later said he “should have used a different example and for that, I apologize,” but stood by his decision not to endorse her.The contest has drawn attention from statewide and national figures as well as a number of Democrats considering runs for higher office.Jumaane D. Williams, the New York City public advocate who formed an exploratory committee for governor, has campaigned for Ms. Walton and urged other Democrats to endorse her, as New York’s U.S. senators have, even as other party leaders have stayed out. Ms. Walton is one of many local candidates who amplified ideas popular with the party’s left — on issues from reallocating funds from the police budget to how best to protect tenants — and won primaries this summer, continuing a trend that began three years ago with the primary victory of Representative Alexandria Ocasio-Cortez, another Walton endorser.“There’s a lot of appetite for these kinds of policies,” Mr. Williams said.The Democratic Party has unquestionably moved to the left in recent years — on issues like criminal justice reform and combating climate change — and Mr. Williams argued that internal divisions are often more a matter of tactics than of substance.“The policies that are being pushed are not really what’s at issue,” he said. “What’s at issue sometimes is how far into political risk, how far past the establishment leaders, how far past, when the executive or leader of the House calls and says no, how far would you push past?”But plainly, there are policy differences among Democrats, too, and in New York those distinctions are especially vivid around matters of public safety. “Do you want to defund the police?” demanded Representative Thomas Suozzi of Long Island, when he campaigned for Mr. Brown in Buffalo.“No!” the crowd replied.“Do you want to let criminals out of jail no matter what they did?” he continued, as the crowd shouted their objection.“We will lose if we let them win,” he said, referencing those who he declared were seeking to push Democrats in an “extreme” direction. “We will lose the American people, we will lose New Yorkers, we will lose Buffalonians if we adopt that type of extremist agenda.”Jesse Myerson, a spokesman for Ms. Walton, rejected the notion that her ideas were extremist, while suggesting that left-wing contenders have been especially successful at energizing voters.The politicians who are “driving new voter registration, the ones driving small-dollar donations, the ones driving more volunteers to knock doors and make calls, you’ll find that they are Bernie Sanders and Alexandria Ocasio-Cortez and Cori Bush,” he said. “And other politicians whose vision closely aligns with India Walton’s, and not the pro-corporate Democrats.”But Mr. Suozzi, a potential candidate for governor next year, argued in an interview that if Ms. Walton wins, “that’s a national story that is bad for Democrats.”Gov. Kathy Hochul, a former congresswoman from Buffalo, has likened herself to President Biden, who won the election as a relatively center-left Democrat.Nicole Craine for The New York TimesMajor 2022 races in New York will also help shape the narrative about the direction of the party. Ms. Hochul, who succeeded Mr. Cuomo after his resignation this summer, is running for a full term. Letitia James, the state attorney general who has closer ties to New York’s institutional left, is challenging her, and others including Bill de Blasio, the New York City mayor, may jump in, too. And a young, diverse class of incoming New York City Council members is preparing to reshape City Hall, with machinations around the council speaker’s race in full bloom.But one of the biggest national stories coming out of New York has involved Mr. Adams, who would be the city’s second Black mayor. He won the primary on the strength of support from working- and middle-class voters of color and declared that America does not want “fancy candidates,” despite his own close ties to major donors.Some national Democrats have embraced him, believing that he offers a template for how to promote both police reform and public safety — though whether that lasts will hinge on how Mr. Adams, who has faced scrutiny over issues of transparency, finances and past inflammatory remarks, governs if he wins.Still, Representative Sean Patrick Maloney of New York, who chairs the House Democratic campaign arm, has described Mr. Adams as “a rock on which I can build a church.” “What Eric Adams’s victory showed me is that the Democratic Party, at its best, is a diverse blue-collar coalition that doesn’t fall victim to elite or academic notions about what makes sense in the real world,” he said.Mr. Adams and Ms. Hochul — a former Buffalo-area congresswoman — have both likened themselves to Mr. Biden.The comparison, allies say, is as much about tone, faith in relationship-building and a sense of pragmatism as it is about a particular policy agenda. But if the two Democrats presumed to be the most powerful leaders in New York are considered relative moderates, that hardly reflects the entirety of New York’s incoming leadership.In New York City, there are signs that the likely next comptroller, some presumptive City Council members, the public advocate and possibly the likely new Manhattan district attorney will be to the left of Mr. Adams on key issues, setting up potential battles over how to create a more equitable education system, the power of the real estate industry and big business, and the role of the police in promoting public safety.Ms. Hochul, for her part, came to office with a reputation as a centrist, but she has pursued a number of policies that have pleased left-wing lawmakers. Rana Abdelhamid, who is challenging Representative Carolyn Maloney, noted that Ms. Hochul has embraced proposals like extending the eviction moratorium — a sign, Ms. Abdelhamid suggested, of the power of the left: “Because of this progressive movement and because of the organizing and because of progressive electeds really gaining momentum.” The race for governor, already underway, will accelerate as soon as Wednesday as the political class heads to a conclave in Puerto Rico. That election will become the next major battle over the Democratic direction, in a midterm year that is historically difficult for the president’s party. But many political leaders say the question is emphatically not whether New York remains a Democratic stronghold — it is about what kind of Democrats win.“It’s going to be either blue or dark blue,” said former Representative Steve Israel of New York. “If you have more Hochuls and Adamses being elected, it’s a lighter shade of blue; if progressives and ‘The Squad’ surge across the state, obviously it’s a deeper blue. The fact is, it remains blue.”Julianne McShane More