The second Oh Baby was written for the 1928 Broadway Musical 'Rain or Shine'. The principal composer of both the words and the music seems to have been Owen Murphy, though Jack Yellen and Milton Ager were also often credited (they were responsible for the music of the entire show and also seem to have run the company that published it).
Donaldson's Oh Baby has to be played briskly. It has a sprightly Verse of 16 bars which should not be omitted: it rattles along and lends itself to some good rhythmic effects. The Chorus that follows comprises 32 bars in an AABA structure. This Chorus has two distinctive features. First, all four of the eights begin with a chord sequence of I : VII7. I can think of no other tune in which all four eights do this. The second distinguishing feature is the use of bars in which the first melody 'note' is a silent crotchet; and this is followed by three sounded crotchets. This first-note-rest happens in no fewer than 14 of the 32 bars, giving syncopated and staccato effects. All this makes it an interesting instrumental number, very good to play. The vocal (with the words matching those rest-crotchet-crotchet-crotchet patterns) begins: Oh Baby! Oh Baby! Don't say 'No'. Say 'Maybe'.That's just as good as 'yes' to me...
You can hear a recording of Murphy's Oh Baby in a fine Ted Lewis version by clicking here, and Donaldson's Oh Baby, played by Bix Beiderbecke and The Wolverines, by clicking here.
By the way, making matters even more confusing, there have been yet more tunes composed since the 1920s with the title Oh Baby!