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]
(..) can't come home. And my daddy's... a lawyer. Aw... Brenda... what if you were engaged to a doctor? Will that change anything? What? What if I went to your parents... (..)2002 Catch Me If You Can