Almost all belted magnum cases are short to the shoulder as they have to be able to fit in a super tight chamber. As they headspace on the belt, this is not an issue unless you reload. The way to avoid this issue with any belted case is to take the following steps.
1. FL size, trim and chamfer NEW cases.
2. Lube the inside of the case neck
3. Run you case into a neck expanding die. In your case (and mine) a 41 magnum pistol die works fine. Don't "bell" the mouth, just use the portion of the expander to open up the whole neck to +-.409"
4. Now get out your rifle.
4. Start running the case into you 375 H&H FL die to reduce the neck back to 375 H&H dimensions. It's a try and reset process. You want to leave a ridge below the neck and above the shoulder that is just enough to let you close the bolt with a positive feel. Now you are headspacing on the ridge, not the belt.
5. Load em up (full loads) and bang away.
6. You will now have cases fitted to your chamber, that headspace on the shouder.
375 H&H cases are very durable and will last forever with reasonable loads.
The same method is not a bad idea on ANY new case belted or not as brass is always made to minimum size whereas chambers vary a lot.
I even do it on my ancient M-1 and the LC NM cases last forever.
Just be sure to function test any ammo you plan to use for hunting or self defense.
As for the brass you have already fired and stretched. Probably okay for cast bullet loads but were it mine, I'd trash it.
There is another way to do this that also works in bolt actions (more camming power) if you have long enough bullets and a short enough throat. Seat the bullets way out so they firmly engage the lands. The cartridge is now headspaced by the bullet, not the belt or shoulder. Use about a 75% load as there is no bullet jump.
Finally either method works fine with cast bullets or pistol bullets. On my 35 Whelens I use the "ridge method" with 158 gr FMJ .357 revolver bullets seated BACKWARDS.
Great gun, great caliber, good shooting !
