posts are interesting, but!
* borrow another seater, and see if you can see a difference. Never ever been fond of Hornaday dies and there seaters are even worse. See if you can borrow a Forster or even a Redding seater. Square up the seater before you start seating bullets. Try another caliber to see if the results are the same.
* What are you checking runout with? The Hornaday gauge is junk. Neco is the best and most accurate you can buy off the shelf. Wand type indicators are much more accurate..
* if nothing improves, the next place to look is the press itself. The shell holder is probably OK if you really are seeing .001" runout in the case necks. My guess is that it's closer to .0035", as it often doubles the runout loaded. Put an indicator on the bushing the die threads into, to see if it's moving under pressure. If it is, then you have a place to start. If the die threads are loose in it, you have another place to work.
If the cases are actually within .001", then the die is suspect. You might check the cases for a banana. This could make the case go into the seater cockeyed.
gary