I wouldn't go to sleep for my nap unless I played that album--40 odd years ago. Definitely from the '60s and not from the Broadway version.
Here are other songs I remember, besides Me & My Shadow:
I can fly, I can fly, it's as easy as pie, I can fly like a bird, I can fly. Over the ? and through the room--(something...)whirling and wheeling and feeling as high as the sky, I can fly, I can fly, I can fly.

Peter Pan, Peter Pan, who are you Peter Pan?
I'm a little boy, who never grew up, my name is Peter Pan. I'm captain of the little boys, in Never Never Never Land.
Peter Pan, Peter Pan of Never Never Never Land.

Peter also said something like: "Never have to go to school, never wear a tie, never wash my face, never comb my hair, never have to do nothin' I don't want to do...

There was a whole section of the record about the crocodile who swallowed a clock--but that part used to scare me, so I skipped it.

Then there was the rhyming part where Wendy sewed on Peter's shadow. Something like... W:I'll sew your shadow on. P: Ouch it hurts, ouch it hurts, please don't take so long. T: Be still, be still I'm almost done...there, your shadow's on. P: Thank you Wendy

Thank you for that trip down memory lane. Then there was the Ugly Duckling album...