In a fusty manner; in a musty, stale, or stuffy way; in an old-fashioned or stubbornly conservative way.
From fusty with the adverbial suffix -ly, creating an adverb that describes how something is done or how someone behaves.
The adverbial form is rarer than the adjective, which tells us that people were more likely to call things 'fusty' than to describe actions as happening 'fustily'—it's harder to do something in a stale way than to be stale yourself.
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