WASHINGTON — For Senator Jeanne Shaheen, a New Hampshire Democrat seeking a third term next year in a traditional swing state, a vote on whether to oust President Trump months before they both face voters would be difficult under any circumstances.

It is all the more daunting for Ms. Shaheen because she might have to do so while facing the closest thing to the president himself: Corey Lewandowski, the combative former Trump campaign manager who hails from New Hampshire and has threatened to challenge her.

“It should be a tough vote for everybody because we should take this issue seriously,” Ms. Shaheen said of a vote to impeach the president. “People ought to be looking at how real the allegations are, whether they are impeachable offenses, and whether the president engaged in them.”

While much attention has been focused on how Senate Republicans will cope with an impeachment trial, the prospect presents its own problems and challenges for Democrats in their uphill fight to win back the Senate majority.