TerryR
As my friend Rusty has stated I have used my doubles for quite a bit of NA hunting.
With my 9,3x74R Chapuis [with iron sights and with a scope] I have taken small game, bobcat,coyote,turkeys,deer, wild pigs, and black bear. I have hunted with it in Texas, Montana, Idaho, and Zimbabwe. I will be taking it to Canada in August for black bear.
With my 450/400 3 1/4" I have taken deer,wild pigs, black bear and caribou. I have hunted with it in Texas, Montana, and Alaska. I hunted with it in Alaska for brown bear also but did not see a "shooter"
With my 450 No2 I have taken deer,and wild pigs in Texas and 2 buff and 3 elephant in Zimbabwe.

I have found the double rifle to be the best "hunting" rifle there is. With the 9,3 [scoped] I have made a 271 yard shot on a coyote and killed a kudu at slightly over 300 yards.
The more you use your double the more you will like it.
I do not know if your 375 is scoped, if not I HIGHLY recommend you get a good scope fitted. A scope on a 375 double would be just about the best all round hunting rifle a fella could have.
What loads do you shoot in your double. I have had excellent results shooting several different bullets in my doubles, most of the time not even having to change the zero of the rifle.