Alexander was born 18 April 1879 in Abbeville in northern Mississippi. He was the eighth of eleven children of the Rev. Duncan Preston and Maria Shaw. Alexander's parents were ex-slaves; his father also an ordained minister in the M.E. Church, as was Alexander's elder brother, J. Beverly F. Shaw. Alexander married Lottye Blanche Simon 29 March 1911. They had children Alexander Preston Jr., Bernard Johnson, twins Lena Anita and Bessie Elaine, Helen Marguerite, and Wilbur Allen.
Shaw was elected to the episcopacy of the Methodist Episcopal Church and consecrated a bishop by the 1936 General Conference. Shaw was assigned to the New Orleans episcopal area, and then to the BaltimoreEpiscopal Area of the Central Jurisdiction of The Methodist Church. As Resident Bishop of the Baltimore Area, he served as the presiding bishop of the Delaware, East Tennessee, North Carolina, and Washington, D.C. Annual Conferences of the Central Jurisdiction. These conferences together included some 1,300 African American Methodist churches and about 100,000 church members. Shaw also served on the so-called "Committee of Twenty-eight", the Board of Education, the Board of Missions, the Board of Temperance, and the Committee on World Peace. His office while bishop of the Baltimore Area was located at 1206 Etting Street, Baltimore 17. His home was at 828 North Carrollton Avenue, Baltimore.
As pastor/preacher
Described by Time Magazine as "tall", Shaw also was well-poised. When he was appointed to Winchester, Virginia, he found the second-floor ceiling of his parsonage too low for him. When he solved this problem by persuading his congregation to rent him another house, while leasing the parsonage "to a much shorter man," newspapers in the Washington and New York areas delightedly picked up the story, causing the Rev. Shaw "a good deal of embarrassment." Nevertheless, Shaw was a brilliant preacher. His specialty was his urgent appeal to youth. Speaking of the Wesley Chapel M. E. Church in Los Angeles, the city's fifth largest, Shaw said, "My Church was not filled with bald-headed people. We had regularly as many as 200 to 300 youngsters attending services." Reportedly, he would occasionally bring an outstanding boy or girl into the pulpit with him, to lecture on how the church could be made even more interesting to young people.
Advocate for his race
Shaw was also reported by Time Magazine as "consistently advocat self-improvement and development for his race." This was distinguished from "the rough, wild way of pressure groups trying to stamp out anti-Negro activities." Shaw held that "a sufficient amount of real excellence – as has been achieved by Marian Anderson and Dr. Carver and Jackie Robinson – is the surest way." Above all, said Time, " believe that Negroes should observe the rule he himself has followed with such conspicuous success: take on responsibilities." Said Shaw:
If you don't have responsibilities, you don't grow strong enough to handle them.
Honorary degrees
Shaw was honored by Philander Smith College in 1916 with the Doctor of Divinity degree. Rust College did the same in 1920, as did Gammon Theological Seminary. Boston University awarded the D.D. in 1937. Rust College also awarded Shaw the LL.D. in 1937.
Retirement, death and burial
Shaw retired in 1952 from the active episcopacy of The Methodist Church. He was, however, called out of retirement in 1953 to again serve part of the New Orleans Area following the death of Bishop Robert Nathaniel Brooks. Shaw served the rest of that quadrennium in this position. Shaw died on March 7, 1966. He is buried in Los Angeles.