This is quite a different song from the one posted by Art Thieme (which he says was written by Jay Stielstra), though it has the same chorus and may be sung to the same tune. I don't know which came first, and I don't know which one Cap't Bob had in mind.

TITTABAWASSEE JANE
As recorded by Blue Water Ramblers on "Coming Home" (2006)

Tittabawassee Jane's her name, Tittabawassee Jane.
The river there must be named after her; she's my Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
The river there must be named after her; she's my Tittabawassee Jane.

Way back in 1891, I came to Michigan.
I's gonna cut some trees but I fell in love with Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
I's gonna cut some trees but I fell in love with Tittabawassee Jane.

Went to town on a Saturday night just to raise a little Cain,
But instead of a fight, it was love at first sight with Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
Instead of a fight, it was love at first sight with Tittabawassee Jane.

Sleep all night and work all day in the sunshine and the rain,
But I didn't mind 'cause I knew I'd find my Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
I didn't mind 'cause I knew I'd find my Tittabawassee Jane.

For two long years I saved my pay, bought a farm with a weather vane,
And I married my gal and settled down with Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
I married my gal and settled down with Tittabawassee Jane.

The years went by, we raised some kids, and now I use a cane,
But when push comes to shove, I'm still in love with Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
When push comes to shove, I'm still in love with Tittabawassee Jane.
Tittabawassee Jane's her name, Tittabawassee Jane.
When push comes to shove, I'm still in love with Tittabawassee Jane.

- - -
Unfortunately, the Blue Water Ramblers' website give no information about the origin of this song.

According to the International Railway and Steam Navigation Guide of 1879, there was once a railway station called Tittabawassee June on the Saginaw Valley & St. Louis Railway, just 4 miles from Saginaw. Coincidence?