Deakin, then get rid of the .300 and use the 30-06 and not worry about it. There are various ways to cure yourself of flinching but they would require a year long regimen of shooting the .300 (actually anyone that shoots a boomer well will tell you that you need to shoot them often to keep your hand in) but IMO you'd be better off spending your time practicing with the '06. You say you shoot it well so why not just get better?
I killed my first elk with a 30-06 and a 200gr Speer Hot Core bullet.
I don't know how you are mounting your .300 but to shoot a boomer, you must handle it firmly. Grasp the wrist with authority and the forearm tightly and pull them into your shoulder firmly! Above all, don't shy away. You don't want the rifle to get a running start. Then the rifle and you move back as a unit.
