Governor Bob Taylor of TN (19th century) was a fiddler and has a tune or two named for him.

Another fiddling politician was Al Gore's dad, Sen Albert Gore (D-TN).

Jimmie Davis, who counts as much a folkie as anybody mentioned so far, was elected Governor of LA, as much for "You Are My Sunshine" as anything else. (One of his opponents commented that Davis would sing "How Great Thou Art" at a church social in northern LA and "Bed Bug Blues" at a barbecue in the south.)

W Lee "Pappy" O'Daniel, governor of and later Senator from TX, promoted himself into politics with a western swing band on radio and records -- originally the Light Crust Doughboys. Bob Wills and Milton Brown both got their start as musicians for O'Daniel and his Light Crust Flour company. O'Daniel wasn't himself a musician.

Lomax's Our Singing Country refers to the banjo picking High Sheriff of Hazard Co [sic!], whose music gets him re-elected every year [sic!]. If memory serves, they're speaking of Justis Begley, who recorded a number of excellent tracks for the LC.

Banjo Bill Cornett, from Knott Co, KY, was elected to the state legislature. His music was no small part of his campaign, and he even composed a song, "Old Age Pension Blues," in re: one of his pet causes, which he rendered with banjo on the floor of the house.

On the other hand, there's also "Bob Roberts." And to go even further to the Dark Side, both Orrin Hatch and John Ashcroft are songwriters. One hesitates to mention Hon. Sonny Bono (R-CA) as a "singer." Mike Curb ran for but lost the governorship of CA.

I'm sure there are others I'm leaving out.