There should be flats milled on the ballscrew... somewhere. Also did you use the provided spacers?
The springiness could be flex in your machine. The stepper is also like a spring, since it's basically a big magnet; you could be back driving the screw. Finally the coupler vould br a source of flex if it's a spring type or Lovejoy with an improper spider. Some couplers are designed for a little lash to absorb shock from the liad or reversing. Using Oldham style couplers can mitigate that movement.