Fuller's Earth does work well for making a dark hat look dusty. I was just working on weathering a Butch Cassidy hat to replicate the one Paul Newman wore in the first half of BUTCH CASSIDY AND THE SUNDANCE KID.
In the past, I've used a wide makeup application brush like this one at Target --
https://shorturl.at/qX4pZ -- which you can find at any pharmacy or the make-up aisle at most grocery stores. I also use a half-inch wide art paint brush for smaller details. I dip them very lightly into the Fuller's Earth and apply sparingly in layers. Quite a few of the grains will fall off, and can be brushed off if they land where you didn't want any dust.
But what actually worked best for application was a soft foam makeup sponge like these --
https://shorturl.at/R3RyY
I also tried something new on the Butch Cassidy hat -- shavings off various earth-tone compressed pastel chalks like this set --
https://shorturl.at/pofOz
I just rub the color chalk sticks against a cosmetic foam-core file and collect the dust in a container. By mixing the colors, you can get a wider variety of colors on the hat.
The chalk dust seems to stick to the hat felt better than Fuller's Earth, which probably means it's tougher to clean off, so keep that in mind. With my Butch Cassidy hat, I wanted the weathering to be closer to permanent anyway.
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