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]
(..) to say this, Lieutenant, so I'm not going to try. Claims have been made that you have, uh, engaged in fraternization of the same sex variety. Specifically that you were... "seen in public venues with another female officer at a time and manner... (..)1997 G.I. Jane