FAIRFAX, Va., Nov 3 (Reuters) - Republicans pushed Democrats out of the Virginia governorship and were running even in heavily Democratic New Jersey on Wednesday, signaling trouble for President Joe Biden’s party heading into next year’s congressional elections. Republican Glenn Youngkin here, a former private equity executive, claimed victory over Democratic former Governor Terry McAuliffe in Tuesday"s vote after distancing himself just enough from former President Donald Trump to win back moderates who had supported Biden just a year ago. In New Jersey, Republican challenger Jack Ciattarelli and incumbent Democrat Phil Murphy were locked in a virtual draw, even though registered Democratic voters outnumber Republicans there by more than 1 million. Still, Democrats clung to hope because more votes were due to be counted in their strongholds. Both saw strong gains in the suburbs from independent voters who had been turned off by Trump’s style of politics. The results in states that Biden won easily in 2020 suggested that Democrats’ razor-thin majorities Congress were highly vulnerable in the 2022 elections. Republican control of both, or even one, chamber of Congress would give the party the ability to block Biden’s legislative agenda during the final two years of his current term in office. The results could also further complicate nL1N2RU0K4 Biden’s hopes of passing twin bills worth a combined $2.75 trillion to rebuild the nation’s roads and bridges, as well as bolster the social safety net and fight climate change. They have already been held up by months of infighting between Democrats’ progressive and moderate wings and the election defeat could leave some moderates less willing to back the big-ticket bills. REPUBLICAN ROADMAP Youngkin, 54, declared victory after a campaign in which he focused on parents’ anger over schools’ handling of COVID-19, as well as teaching on race and gender issues. He walked a fine line on Trump, taking care to not alienate the former president’s hardcore base without offering a full-throated endorsement of his false claims about widespread election fraud. McAuliffe’s efforts to paint his rival, a former chief executive of the Carlyle Group Inc, as a Trump acolyte fell flat with voters at a time when Biden’s approval ratings are at the lowest level of his presidency, according to the latest Reuters/Ipsos national poll, conducted last Wednesday and Thursday. “Together, we will change the trajectory of this commonwealth,” Youngkin told a rally in Chantilly, Virginia, early on Wednesday. Republican congressional campaigns may follow Youngkin’s model of focusing on culture wars and promising to give parents more control over public schools. Youngkin leaned into the Republican Party’s expressions of outrage over the discussion of systemic racism in schools. He vowed to ban the teaching of “critical race theory,” a legal framework that examines how racism shapes U.S. laws and policies, while ignoring the fact that Virginia school officials say the subject is not taught in classrooms. He drew sharp criticism from Democrats when he initially hesitated to denounce Trump’s insistence that the 2020 election was “stolen” from him, false claims that have continued to rile Trump’s supporters and led to a mob of them attacking the U.S. Capitol on Jan. 6. Youngkin said later that Biden had won legitimately, but then called for an audit of Virginia’s voting machines, prompting Democrats to accuse him of validating Trump’s election conspiracy theories. The Republicans also appeared to erase the Democrats’ 10-seat lead in Virginia’s House of Delegates, appearing to gain a 50-50 split or perhaps a one-seat advantage. The Republican candidates for lieutenant governor and state attorney general were also leading their races in Virginia. Virginia Republicans picked Youngkin in an unusual convention format in May, rather than by a statewide primary. That format was designed to pick a more moderate candidate, rather than one more closely allied with Trump. Even so, Trump sought to claim credit, thanking “my BASE” in a statement for putting Youngkin over the top. NEW JERSEY WOBBLES The New Jersey race remained too close to call as dawn approached on Wednesday. But a loss for Democrat Murphy would send even more chills through the Democratic Party, which has been unable to pass Biden’s signature legislation nationally despite razor-thin majorities in both houses of the U.S. Congress. Murphy, 64, ran as unabashed liberal, seeking to become the first Democratic governor to win re-election in New Jersey in four decades. Ciattarelli had faced an uphill battle in New Jersey, where Democrats heavily outnumber Republicans. Ciattarelli, 59, is a former state legislator and business owner who has criticized Murphy for requiring masks in schools and day-care facilities. He campaigned on cutting taxes and supporting law enforcement, but does not support banning abortion -- an unusual position for a Republican. According to local media, Ciatterelli appeared with Trump at a “Stop the Steal” rally in November 2020 in which the then-president falsely claimed to have won the election. Ciatterelli has since said that Biden won the election fairly. Reporting by Joseph Ax, Gabriella Borter and Jason Lange, additional reporting by Andy Sullivan, Kanishka Singh and Daniel Trotta; Editing by Scott Malone and Kim Coghill