Clearly you need to adjust your rail(s). The action was designed for the 8x57. The shoulder on the .338-06 is further back, thus it doesn't kick over as far when feeding. The shoulder on the 9,3x62 is further forward, hits the "bottleneck" in the rails sooner than the 8x57, and kicks the bullet nose over further, hitting the ass end of the barrel. You can screw a barrel on easy enough but you should expect to have to give some attention to the feed rails if you expect proper feeding.
But cheer up, the 9,3x62 is easy enough to make work properly.