If resizing is giving you runout with the buttons removed and the shellholder floating, I'd definitely blame the die.
One other thing to try.
1) loosen the lockring on your sizing die
2) put some up pressure on the die, by placing something between the shellholder and the bottom of the die (I use a small 1/4" shim with a hole through it)
3) while pushing up, lock you lockring.
This takes some of the slop out of the threads between the die and press and has been known to straighten things up a little.
ps: I'd get the Redding dies anyway