I asked Art Thieme if he knew anything about this. He sent me this by email:

Could that be the song called The Browns Ferry Blues? One of the verses starts with those words.

Let it rain,
let it pour,
Let it do a whole lot more (or something like that)
Lord, Lord, I've got the Browns Ferry blues.

I'm going back to Muscle Shoals,
Stay there with my ma and pa,
Lord, Lord, I've got the Brown's Ferry blues.

I hear it done by Grandpa Jones, The Delmore Bros., and Merle Travis---called the Brown's Ferry Four.

But I think the New Lost City Ramblers did it, possibly from Charlie Poole. Doc Watson too.

Art