I found "The Old Music Master" on the Internet Archive among a collection of Paul Whiteman music.

I later found the same version on Youtube where the description says it is Johnny Mercer and Jack Teagarden (which sounds right to me) and that is was recorded on June 12, 1942. One of the comments below says it is the Paul Whiteman Orchestra.

Here I found confirmation that the above info is probably accurate.

I know the phrasing is no longer considered appropriate and I probably wouldn't play it if my son's girlfriend or her dad were around but for myself, I like it. Of course I am partial to Jack Teagarden.

I just found a version by Hoagy Carmichael where it sounds to me like he says 'colered boy' but that doesn't mean he didn't record a version with 'curly boy' instead.