BEAUMONT, Tex. — Operating on two hours’ sleep, Beto O’Rourke was 20 hours into his day and looked it. His white shirt and gray slacks were an accordion of wrinkles. His hair, flecked with gray, drooped on his forehead and small dark rings had formed under his eyes.

But he hadn’t lost his voice. The Democratic congressman from El Paso was speaking to a crowd of several hundred at Suga’s restaurant, 830 miles from home, trying to make an improbable case: that he can defeat Texas’ incumbent Republican senator, Ted Cruz.

Democrats need to pick up two seats in the midterm elections to win control of the Senate, but they also must defend incumbents in 10 states that President Trump won. Mr. Cruz is seen as safer than, say, Dean Heller, Nevada’s Republican senator, or the seats in Arizona and Tennessee that are being vacated by incumbents. And with Democratic money playing defense for incumbents in Missouri, Indiana, West Virginia and elsewhere, Mr. O’Rourke expects no cavalry from Washington to come help him.