OK, now this is just too good a piece of synchronicity.

I came across an old cassette tape my uncle must have made from a 2-LP set, "Songs of the South African Veldt -- José Marais". I learned many Marais and Miranda songs from my mother's LPs, and as soon as I saw #4 in the handwritten list of titles, "Johnny With the Wooden Leg", I remembered "Johnny with the Bandy Legs" from M&M. It's the same tune but somewhat different words, as the title reflects, and the tape has only the refrain sung twice with no verse. (Also, this was a men's chorus with no Miranda.)

For the words of "Johnny", I could remember only the chorus and snatches of lines from the verses. But I realized that the verse tune as I remembered it seemed VERY familiar, but from a different song... but I couldn't place it. I hummed the verse tune to my wife, who ID'ed it immediately as "I was born about a thousand years ago"!

So I checked this site for "Johnny with the Bandy Legs", found it, and listened to the MP3. Not quite the same tune as The Liar Song (as I know it, and which I was pretty much humming), especially in the last two lines of the verse, but IMHO pretty close.

Here's the synchronicity, unless there was a search hit I didn't notice: When I paged back, I hit this thread, "Origins: Born 10,000 Years Ago". But I hadn't typed in anything about that, and I don't see any other mention here of the song I was looking for, by either of its titles!