With just hours to go until Tuesday's "safe harbor" federal law-mandated deadline for states to certify their election results and next Monday's Electoral College voting coming up, time is running out for Trump to find a way to overturn the results of the presidential election as he’s set out to do.

The president held a rally in Georgia -- his first since losing the election -- to campaign for the upcoming Senate runoffs there, races which will determine whether the GOP maintains control, but instead he spent most of his 101 minutes on stage pushing a whirlwind of unsubstantiated claims that the election was stolen.

Trump directly called on GOP Gov. Brian Kemp to "immediately ask for a special session of the legislature" -- which could theoretically set up the process of allowing lawmakers to appoint new electors who would support Trump over Biden despite Biden winning by nearly 12,000 votes in the state.

But Georgia officials have rejected Trump's call.

Kemp, joined by GOP Lt. Gov. Geoff Duncan, said in a statement late Sunday it would be unconstitutional for the legislature to "retroactively change" the process for choosing presidential electors, saying "doing this in order to select a separate slate of presidential electors is not an option that is allowed under state or federal law."

While Kemp and Duncan, along with GOP Secretary of State Brad Raffensperger and officials in his office, have said the only option in disputing the results is through the courts, pro-Trump efforts have ended in at least 38 defeats to date, with only a single court victory -- in a case in Pennsylvania that was ultimately not consequential.

It all comes as the president’s personal attorney tasked with leading his efforts to overturn the election in court, Rudy Giuliani, who has been traveling the country as part of the president's ongoing efforts to dispute election results, remains hospitalized with COVID-19 at MedStar Georgetown University Hospital in Washington since Sunday. The Arizona legislature has closed for the week "out of an abundance of caution" after Giuliani met with roughly a dozen GOP lawmakers.

As deadlines loom, Trump on Monday presents the presidential Medal of Freedom to wrestling legend Dan Gable at noon and meets with Vice President Mike Pence behind closed doors.

And while Trump administration officials vow that with vaccine on the horizon, life will soon return to normal, the Biden team is sounding alarms on its distribution, saying they still don’t have access to the plan to roll it out.

"We have yet to see any kind of detailed plan," Dr. Celine Gounder, an expert on Biden’s COVID-19 advisory board, told "CBS This Morning" Monday after Biden said the same last week.

Pressing forward with the transition despite Trump's roadblocks, Biden and Vice President-elect Harris will receive the President’s Daily Brief and meet with transition advisors on Monday ahead of personally debuting their new health picks, who will inherit leading the country through the ongoing coronavirus pandemic.

The Biden team has announced co-chairs for the Presidential Inaugural Committee with Democratic Rep. James Clyburn of South Carolina, whose endorsement helped put Biden on a path to victory, at the helm to guide the larger committee as they prepare for the event in 44 days.

"This inauguration will show the country and the world something I have always known: we know Joe, and Joe knows us," Clyburn said in a statement through the Biden transition team.

Rep. Cedric Richmond, D-La., Rep. Lisa Blunt Rochester, D-Del., Los Angeles Mayor Eric Garcetti, and Michigan Gov. Gretchen Whitmer will help lead Biden’s team as co-chairs, along with Clyburn who will serve as chair.

“These leaders reflect the strength, spirit, and diversity of America and have always held a steadfast commitment to restoring the soul of the nation, building back the middle class, and unifying the country,” Biden said in a statement Monday morning.

-ABC News Molly Nagle

President-elect Joe Biden unveiled his health team early Monday morning, a slate of seven experts and officials that will lead his administration's response to the ongoing COVID-19 crisis.

California Attorney General and former Rep. Xavier Beccera is nominated as Secretary of Health and Human Services. If confirmed, he would be the first Latino to lead the department.

Vivek Muthy has been nominated to be U.S. Surgeon General, a role he served in during the Obama administration.

Dr. Rochelle Walensky, a leading expert on virus testing, prevention, and treatment, is nominated to serve as director of the Centers for Disease Control and Prevention.

Dr. Marcella Nunez-Smith, one of the country’s foremost experts on health care disparities, will serve as the COVID-19 equity task force chair.

As Biden said last week, Dr. Anthony Fauci will stay on in his current role as as director of the National Institute of Allergy and Infectious Diseases, and will take on the elevated role of Biden's chief medical adviser on the COVID-19 pandemic.

Jeff Zients will serve as coordinator of the COVID-19 response and counselor to the president, and Natalie Quillian will serve as deputy coordinator of the COVID-19 Response.

“This trusted and accomplished team of leaders will bring the highest level of integrity, scientific rigor, and crisis-management experience to one of the toughest challenges America has ever faced — getting the pandemic under control so that the American people can get back to work, back to their lives, and back to their loved ones," Biden said in a statement announcing the picks.

-ABC News John Verhovek and Molly Nagle

From the technical standpoint of a transition of power, the public opinion of congressional Republicans about whether Biden will take office Jan. 20 matters almost not at all.

From the broader standpoint of democracy and faith in elections, it could not hardly matter more at this moment.

The process that so many Republicans have been saying they want to play out is rather thoroughly exhausted already. The next seven days bring significant deadlines, with Tuesday's congressional "safe harbor" deadline for state election results and next Monday's Electoral College voting the most significant.

His tweets and the 101 minutes Trump spent airing grievances and falsehoods in Georgia Saturday night made clear he will pursue his dangerous fictions right through the end of his presidency. The good of the Republican Party -- and, of course, the good of the country -- are, at best, secondary considerations for the president at this point.

Some Republicans are trying to reframe questions about Biden's legitimacy as president-elect as though they are merely asking questions. Their suggestion is that democracy is threatened if people believe election irregularities have been ignored.

But far from being brushed aside, Trump's claims have been rejected in courthouses and state houses -- including by Republicans who have found themselves targeted by the president.

Ignoring that fact only sows more doubt in election integrity and legitimacy. And continuing to give Trump space to make wildly inaccurate and irresponsible claims will only solidify his hold on a party whose best interests have already begun to diverge from the president's.

-ABC News' Political Director Rick Klein

Biden and Harris formally introduced newly nominated and appointed members of their administration from Wilmington, Delaware, Friday afternoon, showing off several familiar faces, be it from the Obama administration or from Capitol Hill.

For director of the White House Domestic Policy Council, a position which does not require Senate confirmation, Biden picked Susan Rice. Rice served as U.S. ambassador to the United Nations and later as national security adviser in the Obama administration.

In her remarks, Rice laid out the path of her ancestors on both sides who overcame adversity -- a paternal great grandfather who was born a slave in South Carolina and maternal grandparents who immigrated from Jamaica with no education and worked to send their children to college, leading to her mother’s role in creating the Pell Grant.

“Now at the foot of yet another bridge between crisis and opportunity, I'm honored and excited to take on this role. Joe Biden and Kamala Harris's vision for our future is expansive but achievable," Rice said, stressing her excitement to serve again and the need to develop policies that break down racial barriers that hold all Americans back.

Rep. Marcia Fudge, who has represented Ohio's 11th Congressional District for the last 12 years has been tapped for secretary of Housing and Urban Development and talked about the importance of housing in America and the need to give people hope amid  America’s ongoing crisis.

“Perhaps most importantly of all, we will help people believe once again that their government cares about them, no matter who they are. That we understand their problems, as the president-elect often recalls his father's words,” she said.

Biden's nominee for secretary of agriculture, Tom Vilsack, seemed to address some of the criticism of his previous tenure at USDA (he held the same Cabinet position he's nominated for throughout the Obama administration) by promising to deal with systemic inequities.

Vilsack also mentioned his support for Rep. Jim Clyburn's 10/20/30 plan, an effort the South Carolina congressman said aims to help counties that had a poverty level of more 20% for more than three decades. Those communities would receive at least 10% of federal funds from a specific program. When asked in 2019 if he considered the measure a form of reparations, Clyburn said the measure "absolutely" is.

Biden's pick for secretary of veterans affairs, Dennis McDonough, former President Barack Obama's White House chief of staff, pledged to make the department more inclusive to all veterans, particularly women, veterans of color and members of the LGBTQ community.

Biden's appointed Katherine Tai to U.S. trade representative, who began by recalling her parents' immigrant story of her father, a researcher at Walter Reed, and her mother, who still works at the National Institutes of Health developing treatments for opioid addictions. She also stressed the role trade can play to create opportunities and lift people out of poverty and the need to rebuild relationships around the globe.

-ABC News' Molly Nagle, John Verhovek and Beatrice Peterson