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]
(..) Miss Scarlett, if true love carries any weight with you... ...you can be sure your sister will be rich in that. I'll go out somewhere and get myself a little business if we're engaged. As soon as I get on my feet again-- All right, Frank. I'm sure I can speak for Pa. You go ask her now. (..)1939 Gone with the Wind