I used Google Book Search to search for "beech oar" and found this, from "Lights and Shadows of American Life" by Mary Russell Mitford, 1832. The chapter is called "The Last of the Boatmen."

As they left the shore, they gave the Indian yell: and broke out into a sort of unconnected chorus, commencing with —

"Hard upon the beech oar!
She moves too slow!
All the way to Shawneetown,
Long while ago."

I also found some quotes--the oldest was 1874--that used the spelling "beach oar." The same 4 lines are quoted in several old books, but none gives more lyrics than that.