From the above, I'll guess you are wet tumbling rather then dry tumbling.Initially, I was tumbling my brass, then loading it. I was having inconsistent primer seeding. More than once the case didn’t pick up a primer. So I just deprimed/sized the brass, then put it back in the tumbler. After that the black crud is gone from the primer pockets.
I use a little Dawn the first go around, then a little brass cleaner the second go around.
I will take a little longer looking at the flash holes while I’m separating the media, that’s a great idea.
You might find it easier/more efficient to use a Universal Depriming die to deprime before tumbling the first time...you won't need a second wash. You don't mention using Lemi-shine, so I'll recommend that...it cuts the carbon. I also use Armorall Wash & Wax in place of Dawn as it leaves a slight protective film on the cases