Isn't Google wonderful!

Fiddlin' John Carson sings Banks of the old Tennessee

And while we're at it, Fiddlin' John sings Sunny Tennessee

I also found a version - possibly minstrel? - on traditionalmusic.co.uk, with the chorus:

"Ho, ho, ho, them banjos,
It would fill your heart with glee,
On a moonlight night, when the stars are shining bright,
'Way down upon the banks of Tennessee."

In the case of 'The Housecarpenter' I suspect that the Tennessee was chosen for its rhymability (is that a word?) rather than the merits of the waterway itself.