Options for sorting old and new versions of colors
I too think you answered yourself partially.

You don't want 6 separate containers for these colors, which seems sensible; but on the other hand you admit yourself you mix some other colors when they are easily distinguished. Which I fully agree on: the first sort you should make is by part type anyway, and only afterward, if you have too much of a part, you may want to split the heap by colors - but not necessarily all colors, since your goal there is just to split a big heap to keep it manageable.

So how do you decide which colors to put together? Again, you hinted at it yourself: colors that are too similar are best separated. So you'll for example have a container with black, yellow and red parts (guess which country I'm from ;-)), and another with dark blue, orange and white.

Another possible option is if there is one color which is predominant: if I have, for example, 1000 1x2 blue bricks, while I have only 500 of 1x2 bricks of all other colors combined, then maybe setting the blue apart make sense. And even then, the best is probably to set 900-950 aside, even in a ziploc bag, and put the remainder with the other colors. The ziplog bag will probably stay unopened until I get a project which requires a massive amount of 1x2 blue bricks. (Actually, it's a real example, I did exactly that, and ended up using the blues for a smooth river)

So what does this leave you with the old/new color issue? Simple: you make two containers, labeled "old" (with Light Grey, Dark Grey, and Brown) and "new" (Light Bluish Grey, Dark Bluish Grey, Reddish Brown). That way, you have no problem knowing if it's old or new, and mixing these three colors shouldn't be a problem.