I haven't found much to answer your question, but thought it may be useful to consolidate what I've found.

Variations in spelling abound, from Waddly Acha to Wadaliacha to Waddle-E-A-Cha, and even Waddley Archer.

There are also movements that go with the song. Here's at least one variation on the movements, via primetroop.org

Action
Slap knees twice, clap hands twice.
Pass the right hand over the left hand twice.
Pass the left hand over the right hand twice.
Touch the nose with the right hand; then, touch the left
shoulder and leave the hand there.
Touch the nose with the left hand; then, touch the right
shoulder and leave the hand there.
Put both hands up in the air, and pinch the fingers together
three times.
Repeat the song and actions several times, a bit faster each
time.

You can also see the sheet music to "Doodle Doo Doo" here by clicking the "download" button