Here is how I process my fired .308 brass.
(1) Tumble in corncob media that is mildly impregnated with "Brasso" metal polish.
(2) Remove brass and shake media loose.
(3) Spray with the same Hornady lube you use, several light coats from 3 directions, including the case mouth.
(a) I use Lee Dies also, and my aim is to reduce wear on the die and eliminate drag on the case mouth that could elongate the brass.
(4) Run the lubbed cases through my resizing die.
(5) Wash the cases in Hot water with a small amount of industrial detergent added. Rinse in cold water.
(a) I've experimented in adding a small amount of vinegar to the wash, and it can boost the detergent and really give the brass a nice sheen!
(6) Spread out on an old towel in the basement or garage for several days.
Now I know step #6 causes there to be a drying period, but I processed a sufficient amount of brass years ago in the first run, and stored it away. That way I'm never in a hurry for any brass, & the drying time is of no consenquence.
"I'm your biggest fan, Ms Blake . . . .
Well, the most heavily armed anyway"