According to census records James Samuel Barrett was b. about 1898, probably Georgia (or Alabama), and Effie Blanche Touchton was born in Georgia. They lived their lives, at least until 1940, in Georgia, and at least six children were born in Georgia, so I'd bet that they were married in Georgia. Before 1952, you have to go to the County Probate Office to get a copy of marriage records.
At the time of the 1920 Census, James and Effie were living in Abba, Irwin County, GA, and James lived in Irwin County in 1900 and 1910. Effie was living in Coffee County, GA, in 1910, and online trees show that her father died there in 1911.
Unless there is a relative somewhere with a copy, I think you'll have to figure out which county, probably Irwin but maybe Coffee, and write for the record.