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]
(..) in San Francisco when he got an offer from ConAgra. He said we could start a life together. So, I followed him. To Nebraska? I thought I'd be engaged by now. No offense. -None taken. -That's all right. When I was , I thought by I'd be married, (..)2009 Up in the Air