Republicans are also focusing on winning battlegrounds like Wisconsin, Florida and Ohio, where their nominees for governor are running more evenly against Democrats. But even in Wisconsin, where Gov. Scott Walker, the G.O.P. incumbent, is battling for a third term, President Trump is coming Wednesday to try to shore up the party’s ticket, including its struggling Senate candidate, Leah Vukmir.

“Governor’s races lead the off-year ticket so they work like a big snowplow, clearing the way for the party offices below them,” said Mike Murphy, a Republican strategist. “If you have a weak snowplow, a lot less gets through behind you”

As the G.O.P. struggles particularly in the Midwest, which President Trump sees as perhaps the most crucial part of his base, Michigan offers a vivid illustration of a crucial political dynamic: While control of Congress and the midterm implications for Mr. Trump consume attention, it is the coattails of Democrats at the top of state tickets that could dramatically reshape American politics after 2018.

In Michigan, which the president narrowly carried in 2016, the Democratic nominee for governor, Gretchen Whitmer, has become one of the year’s breakout sensations after cruising in an easier-than-expected primary and establishing a commanding general election lead. On Friday night, she campaigned in this college town with Senator Bernie Sanders to rally young voters — not because she needed help, but because Michigan Democrats now think they could flip as many as four House seats in the state and win other races to loosen the Republicans’ grip on power here.

Polling by The New York Times Upshot and Siena College found that the Democratic nominee had a lead in one of those House races, while another was effectively tied.