375 H&H recoil
The .375 H&H does recoil, but it is tolerable. A heavier rifle helps, but stock design is an important factor. A stock that has a lot of drop to the butt tends to make the perceived recoil much worse, because the force comes back at an angle to the bore, and the muzzle rises more. A straighter stock will allow the recoil force to come back at a lesser angle, and the muzzle will rise less.
Shooting position also is a factor. Off the bench, the recoil seems much more than if you are shooting off-hand. Longer barrels help a bit, because the powder has more time to burn in the bore, and not give a big muzzle blast. If you reload, going to a faster burning powder will alleviate this, and be easier on your shoulder with approximately the same velocities.
In the recoil calculation formula, the factor for powder is 1.5. That is, a 10 grain increase is one and one half times the factor that a 10 grain bullet weight increase would cause. If you load 70 grains of a faster burning powder into the .375 H&H, instead of 80 grains of a slower burning powder, then the recoil will be much lesser than substituting other components.
You should find that .375s tend to be very accurate rifles. If you can handle the .300 Magnum, you should have no trouble with a properly fitted .375 H&H.