Seth: There is a post in the KIDiddles Discussion Board where someone is seeking the same song and says it was on a "Peter Pan" record she had as a kid in the early 60s.

Her quote is similar to yours: "we're always together as close as can be; we make pictures on the floor and the walls, like rabbits and dogs, but that's not all." But she might know more than she posted. She's looking for the recording, not lyrics.

Googling with the phrase "Me and my shadow, my shadow and me" (in quotes) gets a few hits, but they're only brief quotes, with no explanation.

There are a couple of dozen albums with "Peter Pan" as the title, or part of the title. You can find them at allmusic. Maybe if you browse through the track lists and listen to the sound samples, you will recognize the song you're looking for.

If you come up with more information, please come back here and post it.