I finally found the old 78 recording in the attic above the garage. A good friend had a 78 turntable and recorded it for me in mp3 format.
If anyone is still interested:
(Words in brackets were sung by others)

THE THOUSAND ISLAND SONG
As sung by Arthur Godfrey

Row, row, row your boat, up the river St. Lawrence,
Hollering, hollering, hollering, hollering, hollering for Florence.
I left the one I love on one of the Thousand Islands. (Row, row, row your boat.)
But unfortunately I can't remember which one.
So I row, row, row, row up the river St. Lawrence,
And I'm hollering Florence. Oh, where can you be?
(Where, oh, where, where, oh, where can she be?)
I wouldn't be so lost up here on the Thousand Islands. (Row, row, row your boat.)
If we only had met down on the Isle of Capri.
Now I just can't find the place you marked on the post card.
Call the Navy and Coast Guard.
Oh, where, where can you be? (Where, oh, where, where, oh, where can she be?)
Mustn't take a nap. Gotta watch the map.
That was Island 793,
Row from shore to shore. Knock on every door.
How many Thousand Islands can there be?
I better find you soon 'cause I see a thousand row boats (Row, row your boat.)
And the fellers inside look mighty lonesome to me.
When they row, row, row, row up the River St. Lawrence,
They're all hollering "Florence, oh, where can you be?"
(Row, row, row your boat up the River St. Lawrence
Hollering, hollering, hollering, hollering, hollering for Florence)
I better find you soon 'cause I see a thousand row boats (Row, row, row your boat.)
And the fellers inside look mighty lonesome to me.
When they row, row, row, row up the River St. Lawrence,
They're all hollering "Florence, oh, where can you be?" (Row, row, row your boat.)
Hey Florence, (Row, row your boat.)
Hey Flowie, you're harder to find than Chloe!

Cap't Bob