I would buy another brand of die to start with, like a Redding FL die for starters. Not to get into a 'which die is best' ****ing match, but Lee usually does not get mentioned in 'best' discussions. Lee usually gets mentioned in 'cheapest' discussions.
Also, this was asked earlier in this thread, but will a once fired & twice fired brass fit in the chamber before sizing?
Could this 'bulging' be happening during the FL sizing.
The Redding Competition Shell Holder set allows you to keep your shell holder / die contact cam over, while reducing the amount of shoulder set back in .002" increments, depending on which shell holder you use in the set.