He won't be able to attend because he's 'otherwise engaged'. [=he is busy doing something else]
('US') He is 'engaged' in research. = ('Brit') He is 'engaged' on research.
She 'engaged' him 'in' conversation. [=she started a conversation with him]
(..) - So far, I see nothing strange. - Why? Yeah, you're surrealists! But I'm a normal guy. See, in one life, I'm engaged to marry a woman I love. At least, I think I love her. Christ! I better love her! I'm marrying her! The rhinoceros makes love by mounting the female. (..)2011 Midnight in Paris
(..) - What is this? It's a Valium. It'll make you feel better. - You carry medicine? - No, not normally. It's just since I've been engaged to Inez, I've been having panic attacks, but I'm sure they'll subside after the wedding. I've never heard of Valium. What is this? It's the...pill of the future. (..)2011 Midnight in Paris