I'll throw out my $0.02 as well. First, right amount of Lemishine is important. Too much is worse than too little. Depends a lot I think I your water -- I have tried RO water and regular tap. I just use softened tap now, as its closer in my basement man cave (no trapsing through the kitchen to the RO unit). When I over lemishine, I get dull cases.
Second: I think any detergent dish wash liquid is fine. Nothing about Dawn being special. But, use enough. If there are no suds when you open up to inspect, then you used too little. Kind of duhhh... Hard water and dirty cases require more.
Third: Drying. Compressed air is good. One step further is alcohol. $10 for a gallon at Home Depot. Why consider etoh? Because it will disrupt surface tension and rid the interior of cases of water fast. When I am in a hurry, I shake out the newly tumbled cases, throw them in a small container, cover that with ethanol, then pour out the etoh (it can be reused several times) -- the etoh negates all surface tension. THEN, you blow them out with your air.
Not sure how much I love that last drying step -- I've never trusted it to render the cases dry enough to load immediately. Usually next day. But unless you bake or blow, I think beads of water can persist in cases for many hours or even days after washing. I've looked at trays of brass before that I thought must be dry. And saw beads of water....
Finally, after my initial infatuation with stainless tumbling, I now tumble less and less. Mostly after annealing, where I have used tempilaq. Or if I have come old brass or a mess. But the SS tumbled brass seems less accurate initially than "old brass"..... More and more we read about consistent neck tension being important -- and fowled brass seems to have that more than squeaky clean stuff.
But stainless tumbling has its place, no doubt. You can look inside the brass and see if the flash holes all look nice. Removes all the grunge.....