Why is shady not an adverb?

Shady is an adjective, and the rarely used adverb is shadily.

The Word shady means having or giving shade. You cannot give a physical attribute to a verb (an action).

The slang term shady

If you say that you are "doing something shady" (possibly illegal), it means you are doing something that is shady - it is the noun form of the action that shady modifies.

Example:

The banker was doing something shady with the books.

means

The banker was making a shady change to the books.

The entries he made in his account book were shady entries.

He was engaged in a shady operation.