I've seen the Z done 2 different ways on that machine - judging by your wording I take it someone else converted it and you bought it used? I've seen either the stepper connected to the quill feed handle, or through a ballscrew/ballnut mounted to a clamp attached to the spindle forcing the quill up and down that way. Which way is yours done?
Since you are getting chatter when plunging and noticing digging in I'd first be inclined to say the two issues are connected. Knowing how your Z is driven will help troubleshoot. A pic of your machine would help too.
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