I would like to seriously try and reduce the "felt recoil"

The only way to do that is a muzzle brake. However, both of your rifles are not that easy to do. The double is obvious and the Ruger will have issues with the front sight and band.

Magnaporting could be done to either rifle. Magnaporting creates an illusion for the first couple of shots that the recoil has been reduced a fair bit. However, for hunting that is OK since you are not firing a lot of shots. But remember that once Magnaported you can't unmagnaport unless you cut some off the barrel or barrels

Personally, I think your best bet is to follow some of the advice above and leave the rifles as they are until you have at least played around a bit with them. In general I would put the 30/06 wth max 180 grain loads and the 375 in the same general group for recoil. For sure the 375 kicks more but they are in the same group with the 30/06 at the lower end and the 375 at the upper end.

Over many years I have seen a lot of people take up the 375 and the only ones who had problems were those shooters who did not really like big bores, that is, the simply had more interest in faster an smaller bullets.

It is for good reason that the 375 totally dominates the big bores and recoil is the reason. Once you move much above 375 you are into another recoil class. Recoil above the 375 normally means making some concessions to recoil such as the position the rifle can be shot from etc. or adding a muzzle brake.

Overwhelming opinion and experience strongly suggests that unless someone has some type of injury or disability, then the 375 does not need any recoil reduction methods and can be shot in the same manner as a 30/06 would be used but of course for less shots.

The above is also backed up by what Weatherby sell. The 300 and 340 Wby are in the same class of recoil as the 375 and in standard form Weatherby sells those two without a muzzle brake. However, all of the calibres that are based on the 378 case come standard with muzzle brake.

So in a nutshell the odds and also history are totally on your side in terms of being able to use the 375 without any problems and without having to resort to recoil reducing add ons. Initially, if you have any difficutlies then it should be a matter of looking at how you are shooting or perhaps easing into the guns with lighter loads etc. Again, overwhelming opinion and experience strong suggest that one way or the other you should be able to finish up using the 375 without and recoil reducing add ons.

Mike