I think Brickburn hit the nail on the head with his reply early in this thread. There have been too many guys that have been told or read or think that they need a new "big" gun to hunt Africa, so they show up with a shiny new Weatherby that they haven't shot much, or can't shoot, or are afraid to shoot, and end up not shooting well on their hunt.
Custom, you posted that you like and can shoot your .300 Winchester well. The .300 Win mag can easily kill any of the African plains game animals. Use some of the money you are thinking about spending on a new .375 H&H and buy ammo for your .300 Win and practice, practice, and practice. Buy or make yourself a set of 3 leg shooting sticks and practice shooting off them. Then take your .300 Win and a couple of boxes of 180 gr hunting bullets that are the most accurate in your rifle and have a great hunt in Africa. You need to gain experience hunting in Africa, your rifle doesn't.
Then, after you have experienced your first African hunt, and you want to return for a buffalo hunt, buy yourself a .375 and practice, practice, practice with it for that hunt.
I took a 7mm Rem mag on my first South African plains game hunt and it worked great for all animals including Kudu, Blue Wildbeest, and Gemsbok. For my second African trip, I wanted to hunt buffalo in Zimbabwe, so I built myself a .375 RUM and it worked great Buffalo, Nyala, and a few other animals down to a Chobe bushbuck. Then on my third African hunt I only took my .375 RUM and again it worked great on a variety of plains game from Eland down to Steenbok, and a Vaal Rhebok at 250+ yds. This past summer I took my fancy, custom .300 Weatherby to South Africa, and it made one shot kills on six plains game animals from a Sable down to an African Civet, and a Klipspringer at 314 yds.
My point is, I took a rifle that I was familiar and confident with on my first African hunt, then when I went on an African hunt that needed a larger caliber, I bought my .375 RUM and practiced with it for that hunt.
I handload all of my ammunition, so my costs are only a fraction of factory loads. With the same bullet, it only costs me 10 cents more to load a cartridge for my .300 Weatherby as it does for my .30-06. I also have cast bullet loads for both my .300 Weatherby and my .375 RUM, so I have very cheap and low recoil practice rounds for them.