Of all the candidates on the ballot Tuesday, Mr. Walker of Wisconsin may be the best-known Republican in the country in danger of losing his election. The governor has all but said so himself, repeatedly warning Republicans that they are facing the prospect of a “blue wave in Wisconsin,” and last week suggesting that he may start the general election trailing his opponent. A pair of polls last month showed the little-known Mr. Evers leading Mr. Walker.

Mr. Evers, the Wisconsin schools superintendent who was the best-known and best-funded candidate in the Democratic primary for governor, prevailed in a crowded field of seven other entrants. Though he will face one of the most favorable national environments for Democrats in more than a decade, he is not a widely known quantity and will have to unite a fractious progressive base in a governor’s race that both political parties see as a defining political moment for the presidential swing state.

But the November election is likely to be a referendum on the polarizing Republican incumbent. Seeking a third term, and what would be his fourth statewide win in eight years, the 50-year-old Mr. Walker is hoping to extend his political life following a presidential bid that faltered quickly in 2016. His message of beating back attacks from labor unions and other powerful liberal interests failed to break through in an election that Mr. Trump turned into a repudiation of conventional Republican politics.

Elsewhere Tuesday, Vermont Democrats also nominated Christine Hallquist, a longtime energy executive who could become the nation’s first transgender governor. She will face the Republican incumbent, Gov. Phil Scott. Ms. Hallquist, as the chief executive of the Vermont Electric Cooperative for 12 years, led a turnaround of the financially troubled utility, and as a candidate she ran on a progressive message that included a higher minimum wage and “Medicare-forall.”

[Read more about Ms. Hallquist’s win here.]

In other trailblazing wins, Ilhan Omar, a Minnesota state lawmaker, is poised to be one of the first Muslim women elected to Congress, winning the Democratic nomination in a left-leaning district. (In Michigan, Rashida Tlaib, who is also Muslim, is in position to win a House seat.) And Jahana Hayes, a former national Teacher of the Year who would be Connecticut’s first black woman in Congress, easily defeated Mary Glassman in a Democratic primary for an open House seat that became something of a proxy battle between moderates and liberals.

Mr. Sanders easily won Vermont’s Democratic primary for Senate on Tuesday, all but guaranteeing his re-election in November. But he is expected to snub the party that he sought to represent in the 2016 presidential election: A self-described democratic socialist, he plans to reject the nomination and run instead as an independent, according to advisers, as he did in his Senate races in 2006 and 2012. He seeks the party’s nomination in order to block any rival from winning it, but then turns it down to protect the image of independence that he cherishes.