Recently, I was recounting to my wife various songs that my grandmother used to sing, because I'm trying to ensure they get passed on to our child, and one of them popped up that I have not been able to track down. My grandmother has since passed, and my grandfather does not remember the song.
The central premise is that a girl is offered a coat twice by different men. Both times, the exchange is something like the following:
Man: "Are you cold?"
Woman: "'Bout to freeze."
Man: "Would you like my coat?"
Woman: "Just the sleeves."
Man: "Full or empty?"
With the first man (that she doesn't desire), she tells him "Empty please", indicating that she's just going to drape the coat over her shoulders, tying the sleeves over her neck to hold it in place. With the second man (that she does desire), she tells him "Full please", indicating that she wants him to hold her.
My grandmother was born in the 1920s, and served in World War II. She was born in Pennsylvania (I'm not certain exactly where), and moved many places before settling down in Kentucky. I'm not certain where or when she learned the song, but I remember her singing it to me when I was a young child, around 1980 to 1990. Probably not actually useful for identifying the song (although maybe indicative if it was trying to replicate a particular performer's voice), but she always did the girl's voice in a high and squeaky tone to differentiate it from the man's voice, which was done in her regular voice.
In retrospect, it might have been more "sing-song" than an actual song, but in my mind, it's a song she sung.