One old fashioned way to correct the fit of the seating stem and bullet is wax. A hard wax, like sealing wax. put the softened wax into the stem, and insert the desired bullet, the wax will harden, forming a good fit to the bullet.
It won't last forever, but can last a surprisingly long (# of rounds) time.
A well built die should be smooth enough inside that a slightly tipped bullet will be straightened without shaving any lead. I don't use Lee dies much, RCBS & Lyman dies that I have will do this, if you use just a little bit of care about the "feel" of the press as you seat the bullets.
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All else being equal (and it almost never is) bigger bullets tend to work better.
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