This is a bit of a stretch, perhaps, but could Oscar Brand have included this in one of his many albums? I have one LP of his called "Rollicking Sea Shanties" somewhere in storage. I cannot recall whether "Shallow Brown" was included, however.

Could the name "Shallow" refer to skin color, much as the term "yellow" was used, as in "The Yellow Rose of Texas?" Perhaps a student of regional vernacular of the time could shed light on this.