In a private meeting on Friday morning, Democratic governors reviewed their political strategy for the midterms: Corey Platt, a strategist for the governors, laid out a map of 18 Republican-controlled states where Democrats could conceivably take power, including five important Midwestern battlegrounds and traditionally conservative-leaning states like Tennessee and Georgia.

Mr. Platt also highlighted several important races where the Republican Party’s rightward lurch had widened political openings for Democrats, people familiar with the presentation said. In Florida, the president’s support for Representative Ron DeSantis, a hard-line member of the House who appears frequently on Fox News, in a contested governor’s primary has badly undermined Republican chances there, Mr. Platt argued.

Mr. Platt also pointed to Adam Laxalt, Nevada’s attorney general and Republican nominee for governor, as a candidate who could be too conservative to win a state Hillary Clinton carried two years ago.

Mr. Laxalt is hindered by the refusal of Gov. Brian Sandoval — a more moderate Republican, who is departing — to endorse him. In a sign that the chill between the two has not thawed, Mr. Sandoval said in an interview here that he “won’t support a candidate that is going to undo anything that I put forward.”

As Democrats weighed how best to harness the anti-Trump energy, Republicans locked in competitive races acknowledged the headwinds they are facing and vowed to emphasize their own governing records and political identities to separate themselves from the president.

Deploying euphemisms about “Washington” or “national politics” so as not to offend the president and his supporters, Republican governors said they would seek to localize their races.

“There’s energy on the left, there’s anger on the left and there’s some signs of organization,” said Gov. Doug Ducey of Arizona, who is facing a competitive re-election bid. “So I think the role of the campaign is not only to communicate the message but to separate from Washington and differentiate what we’ve done over the last four years.”