Baxter was the son of Batsell Baxter, another influential leader within the Churches of Christ. The younger Baxter started preaching in Nashville, Tennessee at the age of 17. He received his post-secondary education at David Lipscomb College, Abilene Christian College (B.A.), University of Southern California (M.A., Ph.D.), and Vanderbilt University (B.D.).
When he returned to Nashville from California, he was appointed head of the Speech Department at David Lipscomb College. Later he headed the Bible department at David Lipscomb College when his father died.
Baxter preached in Nashville at the Trinity Lane Church of Christ from 1946 to 1951. He also preached at the Hillsboro Church of Christ in Nashville for 29 years from 1951 to 1980 when he retired. He was also a writer for the Gospel Advocate and 20th-Century Christian. He wrote a number of books including "The Family of God," "Speaking for the Master," and "If I Be Lifted Up." He also was the first preacher of the "Herald of Truth" television program in August, 1959. When the program went to a radio series, he was featured periodically.