You can hear a recording of MY BABY JUST CARES FOR ME at The Red Hot Jazz Archive, performed by Isham Jones and His Orchestra, recorded September 12, 1930. The vocalist sings only Refrain 1, as posted by Joe Offer above.

I think it was common in those days for the recorded version of a song to omit many of the lyrics that appeared in the printed version. First, they had to make it all fit on one side of a 78-rpm record, which limited them to about 2 1/2 minutes. Second, the big bands of that era gave greater emphasis to the instrumental performance. The vocalist got second billing after the bandleader, and the vocal part often didn't start until the middle of the recording. Sometimes that left time for only one chorus to be sung.

I don't know, but I suppose all the verses and choruses were sung when the band performed live or on the radio. Anyone know for sure?