Not adult literacy, but as a previous challenge was to learn to speak in public, this is to write for the public.
This is another challenge for lawyers in particular. We are used to drafting formal documents, and while Latin is in decline, jargon is not (and it can have a legitimate place).
Legal writing generally requires accuracy and/or the development of a logical argument. Writing for the public generally needs to be simpler (often at 6th form level) and needs to both obtain and keep the reader’s attention. Short sentences work, rather than using subordinate clauses (think Denning). If you can, tell a story.
Know your audience, and pitch accordingly. Have some idea of their level of knowledge of the subject matter. It is patronising to get this wrong, in either direction.
So practise, and ask for feedback. Hone your skills for different purposes.