In 'The Universal Songster', II, p. 332, 1826, "Betsy Baker" is attributed Hudson, and the tune direction is given as "Head Man at Mrs. Grundy's", which I don't know. There is an incomplete copy of the song with music, both from a manuscript, in Randolph's 'Ozark Folksongs' I, #117, Version A. Another with music is said in Steve Roud's folk song index to be in Roy Palmer's 'Love is Pleasing'.

I've never seen it connected to "Yankee Doodle", but so much published about "Yankee Doodle" is pure crap that I wouldn't be surprised about much of anything said in connection with it. The tune below doesn't sound much like "Yankee Doodle" to me, and I think "Yankee Doodle" is about half of a century older than "Betsy Baker".

X:1
T:Betsy Baker
S:Ozark Folksongs, I, #117, Version A
Q:1/4=160
L:1/8
M:2/4
K:D
d|A F F D|C E E G|F D F A|d2d d|\
A F F D|C E E A|d c B ^A|B2B c|\
d e f d|c d e3/2 c/|d e f d|(cd) .e c|\
d e f d|c3/2 d/ e3/2 f/|d c B A|d2 d d|\
A F F D|C E E G|F D A3/2 F/|D2D||]