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]
(..) l couldn't even say what that wss. We didn't exchange two personal words and we've been together through some terrible things. Some terrible things. He wss engaged to a girl in the village. l mean... And us he never once... He didn't ssk me if l could spin the ball at cricket or kama sutra or... (..)1996 The English Patient