"Cool Water," Bob Nolan, was recorded August 8, 1945, on DSVB1110 (Victor 20-1724).
It does include the verse with "Dan'd feet are sore,..."
The film may have left out the verse (too long ago to recall).

Personnel of Sons of the Pioneer-
Bob Nolan, Tim Spencer, Ken Carson(v), Hugh Farr(v, vin), R. McCall(vin), C. Roberts(sg), Perry Botkin(g, ldr), Karl Farr(g), Shug Fisher, G. Braunsdorf (b).

Cool Water-
Bob Nolan(v), Leonard Slye (Roy Rogers)(v, g), Lloyd Perryman(v, b), Hugh Farr(vin), Karl Farr(g).
ldr-leader; b- bass, d- drums, g- guitar, v- vocal, vin- violin, sg- steel guitar.