I have found the best dual purpose powder for both the .338 and the 30-06 to be Vihtavouri N160, if you can find it. It's burning rate is right in between 4350 and 4831, very similar to Re 19, which is also an excellent dual purpose powder. 4350 should work well for both, as will 4831.
Of course, what turns out to be the best powder for one particular caliber may not be the best for the other. If you use different weight bullets for the same caliber, you may find two different powders that work best that way also.
Fortunately for me, for 210, 225, and 250 grain bullets in my .338, N160 works best. It is also the best powder for my '06 with 180 grain bullets.
One word of advice, before you go nuts with powders. You currently have 4350. That's one of the most versatile powders going. If I were you, I would probably get a pound of slower burning powder to try in the .338, like RE 22, N160, or 4831. With the 4350 and one of those, I can almost guarantee you'll find the right combination of velocity and accuracy for just about any weight bullet you'd want to try in either caliber.