Some strange effect while my puppet walking...

Try putting the yellow mesh on in the scene window, then watching the mesh closely. Is it the shoulders jumping around and arms following? Or the shoulders are fine and the hands have the shakes etc. Or elbows? Zoom in on that part of puppet to make it easier to see.

What i would do then is trying adjusting strength settings in behavior panel and handle positions relative to each other. Do you have any dangles? Maybe fiddle with gravity too? i don't know the reason, but feel free to share a screenshot of rigging hierarchy here. I have seen lots of walks without the hand shaking, so i am pretty sure it will be solvable.