"Illustrator" isn't normally a title in the same way that "Mister" or "Captain" or "President" are, although I suppose you could force it into being one, in which case I would put a capital letter on it, but it wouldn't be standard English. Of the alternatives, I like the second one below, although all are fine, if a bit wordy. When the illustrator, known as Adams, lost his voice... When Adams, the illustrator, lost his voice ... Adams was an illustrator who lost his voice, thereby baffling the doctors. I hope this addresses your question.

An illustrator is most commonly known as a job where someone draws pictures for books. So possibly this guy, Adams, is an illustrator? For example, an author is someone who writes books, so if Adams was an author it would read instead in a news article like: "When author Adams lost his voice, his doctors were baffled."

The illustrator's name is Adams