I clean/dust fabric lampshades in our house by vigorously brushing the. with a paint brush. Don' see why that wouldn't help here, or that it could hurt any either.
Just hold it against something stiff like a book or board and brush away.
As for dry cleaning, if you should try that route then try only one first, perhaps the worst, and see how it goes. You could also ask the dry cleaner to test one spot first to see what happens.
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