I went to SA in 2010 for plains game...
I took a 375 H&H shooting 270gr Barnes tsx..
My buddy took his '06 and killed everything he shot at with one bullet each...
I was glad I had the .375 when I shot my zebra, but overall it was probably too much rifle, although I didn't have to chase any animals, especially the steinbok.
Pick a good bullet and learn to shoot your 7mm whizzum off of sticks. Practice at distances up to 200 yards. You will be fine.
