The 'recoil problem' is due to the fact that the 338 has higher "recoil velocity" than the 375 even though the 375 has higher "recoil energy". A fast push rather than a slow push .
Increasing the weight and using a recoil absorbing recoil pad will help that .
You could , if you're a handloader , reduce the velocity a bit.
Many seem to be using a premium 210 or 225 for everything.
And Watson , bring your revolver !