I checked Goodspeed's History of 30 East Tennessee Counties and History of Sweetwater Valley, and the only mention of a Boyd is that pre war their was a Sweetwater business: Boyd, Spillman and Vaughn.
There was no enumeration of Mayors.
I have found this piece which may be of help: "An election was held in 1875 to incorporate the present-day City of Sweetwater. The first Mayor was J.W. Clark."
George Martin