The .338 WM is a cartridge people love to hate. Before I got one 10 years ago, I thought it was the meanest kicker out there, certainly worse than a .375. Until I fired my first shot. I guess if someone does a "try this" at the range, and you're not expecting it, you might get that impression. Oh well.
The .375 H&H is a grand old round, and a good one. But, it's a magnum action. More than that, it's always the first rung on "safari grade" rifles, meaning heavier and more expensive. Weight tames some recoil, I'm sure. Weight also tames the hunter down. My .338 weighs 8 lb with its scope, and I had a wide choice of "standard" hunting rifles from all the big makes.
The .375 has pretty good long range ballistics, but the .338 shines. Not as good as the .300 WM for "flat", but high BC 250s have a usable trajectory and deliver a huge punch downrange. That's really the point of the .338: Not quite as hard hitting up close as the .375 nor as "flat" at long range as the .300, but pretty good at both. If you can handle the recoil (you can if you want to) it's a very versatile hunter. I'm sure your Sako is a good one. I have a Savage "Weather Warrior". Not CRF, but a pretty good action and very accurate.
Now having said all that, have you thought of giving up the >300 yard shots and going with a 9.3x62?