“Do I think it could be easier? Yes,” Mr. Alexander said. “Is the environment difficult to work with? Sometimes it is. But I get a lot done, and that requires working with people across the aisle. It takes a lot of patience and it takes people who trust you and you have to trust them, and sometimes you don’t succeed but often you do.”

“I will be 80 years old when I leave,” he said. “I will have served longer as senator and governor than anyone else in Tennessee history, and I think that’s plenty.”

At 70, Mr. Udall is far from old in Senate terms. But he viewed another Senate run as an eight-year commitment, given the need to aggressively campaign, and he did not consider that the best use of his time.

Mr. Udall is an ardent Democrat but, like Mr. Alexander, he has a history and reputation for being able to work in a bipartisan fashion, particularly on environmental issues. In 2016, he defied expectations and struck a deal with David Vitter, then a Republican senator from Louisiana, to renew and update the Toxic Substances Control Act for the first time in 20 years. Yet even that achievement has led to disappointment.

“It is not being implemented,” Mr. Udall said. “The parts of it that protect the public and protect the food are not being put in place.” He blamed the Trump administration and officials at the Environmental Protection Agency for failing to follow through.

“They are chasing off all these people, the scientists and everyone else,” he said of the current administration. “You could have the greatest law in the world but you don’t get it done.”

Mr. Udall has also made restoring Congress’s sole authority to declare war and bringing to a close what he describes as the nation’s “endless wars” a priority, but he has had little success. Last Wednesday, he lost a vote in the Foreign Relations Committee to prohibit any military operations against Iran that were not specifically approved by Congress. The outcome left Mr. Udall fearing that with tensions rising, the United States could conceivably be in a conflict in Iran before Congress returned from its Memorial Day recess to weigh in.