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]
(..) and I know it now more than ever. - I'm sorry. - You don't have to apologize to me. I'm the luckiest person you'll find today. No, I'm sorry I'm engaged to be married. But you're wrong. I do know you. At least by reputation. Edward Bloom from Ashton. See, I'm actually engaged (..)2003 Big Fish
(..) I've never seen any. That's because we didn't have a proper wedding. Your mother-in-law was never supposed to marry me. - She was engaged to somebody else. - I never knew. Will never told you? Probably just as well. He would have told it wrong, anyway. All the facts, none of the flavor. (..)2003 Big Fish