Yes you have a noise problem, either from poor Grounding or shielded cable not being correctly Grounded
When you say you have a separate supply for the spindle, you can not use a separate supply as this will give you a Ground loop your machine can only have ( 1 ) power supply source
So what cable are you using from the VFD Drive to the spindle???
Have you checked the 4th pin in the spindle plug to see it it has continuity to the spindle body, most don't and this need to be corrected ???
Need to see photos to be able to help get you going