Donald J. Trump’s intensifying battle with his own party is tearing open the nation’s political map, pulling Republicans across the country into a self-destructive feud that could imperil dozens of lawmakers in Congress and potentially throw conservative-leaning states into Hillary Clinton’s column.

Democrats are moving swiftly to exploit Mr. Trump’s crumbling position in the presidential race, aiming to run up a big margin of victory for Mrs. Clinton and extend their political advantage into the congressional elections next month.

Mrs. Clinton’s campaign has concluded that at least two traditionally Republican states, Georgia and Arizona, are realistic targets for her campaign to win over. And Republican polling has found that Mr. Trump is at dire risk of losing Georgia, according to people briefed on the polls, who spoke on the condition of anonymity.

Mrs. Clinton now holds such a strong upper hand that Priorities USA, a “super PAC” backing her campaign, may direct some of its war chest into Senate races, two people said, and may begin broadcasting ads for those contests as soon as next week. Congressional Democrats also hope to persuade Mrs. Clinton to continue pouring money and campaign resources into states like Virginia and Colorado, where they believe her victory is assured, in order to lift other Democratic candidates.