Adams was born in Atlanta, Georgia and attended the public schools in Portland, Oregon. He attended the University of Washington at Seattle, graduated in 1949, and was admitted to Harvard Law School, where he earned his law degree in 1952. Adams served in the U.S. Navy from 1944 to 1946, and was admitted to the Washington state bar in 1952, opening a private practice in Seattle. Adams taught law at the American Institute of Banking from 1954 to 1960, and served as United States Attorney for the Western District of Washington from 1961 to 1964.

Adams was elected as a Democrat to the House and served six terms beginning January 3, 1965. He was chairman for the newly created Budget Committee during the 94th Congress, and was considered to be a strong candidate for Speaker of the House. On January 22, 1977, Adams resigned to become the fifth Secretary of Transportation following his appointment by President Jimmy Carter from 1977 to 1979 and confirmation by the Senate. After resigning his Cabinet post on July 20, 1979, Adams resumed law practice, this time in Washington, D.C., where he was a lobbyist for CSX Corporation and other railroad carriers.

On November 4, 1986, Adams was elected as a U.S. Senator, defeating incumbent Slade Gorton (677,471 to 650,931, 51.2 percent to 48.8 percent). Serving one term from January 3, 1987, to January 3, 1993, he chose not to be a candidate for reelection in 1992 after eight women made statements to The Seattle Times alleging that Adams had committed various acts of sexual misconduct, ranging from sexual harassment to rape. Adams denied the allegations, but his popularity statewide was weakened considerably from the scandal and he chose to retire rather than risk losing the seat for his party. Adams never lost an election, and lived in Stevensville, Maryland until his death due to complications from Parkinson's disease.

Adams was a member of the American Bar Association and Phi Beta Kappa Society. He was an Episcopalian.