I just got back a few weeks ago from South Africa and Barnes 300gn TSX was my choice of ammo in .375 H&H. They will work just fine on your buffalo and plains game. I would recommend that you get some real practice shooting out to at least 250 yards and understand the bullet drop from your rifle and scope set up. I shot one buffalo at over 200 yards, I was glad that I knew the real bullet drop that was going to happen on that shot or I would not have taken the shot.
My son and I both packed Blaser R8's in .375 with spare .300 WM barrels, we never shot the .300 WM and only shot the .375 with the Barnes 300gn TSX.
We also packed far too much clothing, the lady in camp turned laundry around in about 6 hours during the day, we could have taken half of what we packed, it's good to live and learn. The only problem next year could be the big double rifle that I see in my future. We were on a buffalo cow cull hunt, the .375 H&H did very well with one shot kills but some of the bull buffalo I saw were much bigger!!
Good luck on your hunt.