William Adams Dawson was born on 5 November 1903 in Layton, Davis County, Utah, where he attended local public schools. He graduated from the University of Utah Law School in 1926 and commenced legal practice in Salt Lake City the same year. He served as County Attorney of Davis County from 1927 to 1928 when he led the fight against pari-mutuel horse race betting in the county.

From 1935 to 1939 Dawson filled two terms as mayor of Layton. During those years, he was president of the town board of Layton, president of the Davis County Bar Association, Republican party county chairman, and bishop of the Layton LDS Ward. He was a member of the Utah State Senate from 1940 to 1944.

Elected in 1946 as a Republican, Dawson served in the United States House of Representatives from 1947 to 1949. He sponsored bills to provide for multiple mineral development of public lands, to permit public schools to gain title to revenue from public lands, to increase appropriations to complete federal land surveys in Utah, and to authorize the $1 billion Upper Colorado River Storage Project. In Congress, he served on the House Interior Committee and was names to the Republican Policy Committee. He co-chaired a special committee to investigate lead-zinc mining needs and served as a member of a committee on coal research and on another on water resources.

Following an unsuccessful bid for reelection in 1948, Dawson was again elected to Congress in 1952 and served from 1953 to 1959, he was appointed president of the Bank of Kearns, which later merged with Zions First National Bank in 1962, and he held the position of vice-president of Zions First National Bank until 1969.

William Dawson married Gladys Norton in 1927 and was the father of two daughters. He died in Salt Lake City on 7 November 1981 and was buried in the Kaysville Cemetery.