BondoBob,
I expect the M-die to fix the issue for you. Per the article 74A95 linked to, you need to place the bullets in and see that they are upright, and not just drop them in place, but you will get used to the feel of it rapidly. You will get a much more even bulge all around the case (though runout in neck wall thickness will still make it a little uneven in some cases). You can also look at range pick-up brass with calipers. A thinner neck wall (Remington is thinner in 45 Auto and may also be thinner in 380; you can check) will be less prone to producing an interference fit with a tilting bullet, but it also work-hardens from resizing more because it can expand more in the chamber, so it may not last as many reloadings.
74A95,
That article echoes my experience, though mine is with 45 Auto. I could never see an alignment effect on accuracy with jacketed bullets, but with soft swaged lead bullets, getting them aligned with the bore by seating them out to headspace on the throat reduced groups by about 40% and, as a bonus, greatly mitigated leading. Jacketed bullets seem to be tough enough to avoid the extreme distortion the softer bullets have scraping against the mouth of the throat and can straighten themselves on the way into the bore without distortion issues of any significance.
I think the author of the article was hoping that what applied to rifle bullets would apply to pistol bullets, but a little math shows it won't work out for several reasons. One is that rifle bullets generally have a longer aspect ratio than pistol bullets, especially in the ogive. A typical 7.62 ball bullet is almost 3 times longer in calibers than a typical 45 HP. That moves the center of gravity of the bullet further forward of the geometric center of the bullet bearing surface by a factor of that length ratio. A given in-bore-tilt angle around the bearing surface center of that rifle bullet produces 3 times the eccentricity of the bullet center of mass's spin around the bore axis of an equally tilted HP pistol bullet. As a result, the lateral jump caused by that eccentric spin is three times greater for the rifle bullet, even if you load it down to the pistol bullet's velocity in a barrel with the same rifling pitch. But that's not normally what you have. Normally the rifle barrel pitch is faster and its velocity greater. That means the rifle bullet is generally spinning faster than a pistol bullet. At exit from the muzzle, a 7.62 ball round from a 10" or 12" pitch barrel is spinning roughly four or five times faster than a round of 45 hardball from a standard 16" pitch barrel. That increases the speed of the lateral jump and its resulting lateral drift that stays with the bullet to the target. (This effect is the principal reason for not using an unnecessarily fast rifling twist in a rifle.) When the two differences are taken together, even if you assume the same time of flight to the target for the drift to work, the rifle is typically going to exhibit around a factor of 12 to 15 greater drift from bullet tilt than that pistol bullet does. If a tilted 30 cal rifle bullet can produce up to 1 moa of group growth, as A.A. Abbatiello, an Oak Ridge National Laboratories engineer found in his early 1960s study of Lake City NM ammunition, then the 45 will produce about 0.07 to 0.1 moa of difference from that source of error, which, for statistical reasons, is very hard to discern.
The reason the author's tilted bullets actually shot better in that article's test is something I can only make speculative guesses about. Some of the tilted round may have randomly centered their cartridges in the chamber better. No other reason occurs to me at the moment.
I've never had a Ransom Rest produce pistol groups as tight as I could get off bags by hand. I believe this is mainly because it registers on the grip frame of a gun, where a shooter registers on sight alignment. If a barrel fits up into the slide well, sight and barrel alignment should be consistent, but if the slide is not fit to the frame, registering on the grip frame still allows alignment of the bore and target to shift around by as much as the slide and frame can shift. I've seen it suggested that spring-loaded plastic rollers bearing on the side of a slide might help the Ransom Rest maintain alignment better, but I don't know anyone who has tried it.
