When President Trump met in the Oval Office with Ken Paxton on Tuesday, the newly minted Republican nominee for Senate in Texas, he brought handouts.

Democrats may feel bullish about their prospects in the state, but Mr. Trump has been dismissive of James Talarico, the 37-year-old Democratic nominee. Scattered across the Resolute Desk were printed photos of Mr. Talarico, along with an image of the 20th-century cartoon character that the president has compared him to: Alfred E. Neuman of Mad Magazine.

Mr. Trump had personally colored in one of Mr. Talarico’s teeth to create a similar gap-toothed smile, according to a person with knowledge of the president’s rendering who spoke anonymously to describe a private meeting.

If, in Talarico versus Paxton, the Democrats believe they’ve found a rare matchup to give the party a shot in November, Republicans believe they have an opponent they can cast as a wacky liberal from Austin.

With five months to go until Election Day, that same dynamic is playing out across the Senate landscape. Democrats are confident that they’ve recruited the right candidates and that political conditions give them a shot to flip at least four G.O.P.-held seats, which they need to take the Senate majority. Republicans are confident that Democrats will lose anyway, with plans to cast them as unelectable in red states.

Hours after Mr. Trump showed off his drawing, the contours of the Senate battleground map came into sharper focus on Tuesday night with the victory of Josh Turek, a Paralympic gold medalist, in Iowa’s Democratic Senate primary.