Character shifting x position whilst still with idle motion library animation

I am unable to reproduce your original sliding issue. The feet do look distorted, but the behaviors are acting on legs fine. This is how it looks like on my end:

Since the legs are at least reacting to the body movement, the basic behavior setup is fine.

To deal with the distortion, you need to play around with the handle placement, in particular the placement of heels, toes, and stick handles. Here is my tweaked result:

And here is how my right leg handle placement looks like:

Tips

1. Do not expect "Fixed" to lock the handle to a global position. It is to prevent deformation within the layer. In your case, I would take out all "Fixed" from feet
2. Try extending stick handles as much as you can through body parts. Try placing horizontal stick handles at the bottom of feet.
3. Place the heel and the toe at a near-even horizontal level
4. Try placing body-tagged handles at anatomically-correct locations. I tweaked the hips placement in your puppet to get a better upper leg deformation
5. Make sure your main body mesh is not attached with "Weld" but set to "Free" - in your case, you should change "Left Quarter," "Frontal," and "Right Quarter" layers' attach style from "Weld" to "Free"