You're greatly overestimating the problem. Throwing your clothes in the wash? The hell are you doing, emptying the tumbler out over your head?
*If you're pouring the whole thing out, don't. Get a kitty-litter scooper, a deep one, to get the brass out of the tumbler. You can either shake them out one at a time by hand, or use a media separator (worth it).
*Cut down the dust by adding 2-4 tablespoons of mineral spirits to the media. While you're at it, a dab of Nu Finish shines up brass really nice.
*De-dust your media a little bit by running some
used dryer sheets in them.
*Get some powder-free nitrile gloves for reloading. Handy for gun-cleaning, too. Gets you out of a lot of hand-washing.