No, the book gave no tune. If memory serves, there may have been several different variations of "Old Tar River" throughout the book. The lyrics seem to have floated around, probably as per different performer's renditions. In other words, this was not the version you might find on sheet music or a broadside, but rather what was being sung around that time that someone captured.
The name of the book is NEGRO SINGER'S OWN BOOK. It has no date or author, but based on the stuff it includes, it seems like early-mid 1840s. No tunes in it. Just a jumble of song lyrics from different sources. I saw a microfilm of it that I ordered on an interlibrary loan. I was able to copy a few selections, but not all of it. I think I took down the above lyrics by hand.