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]
(..) - Who was that? - Lance Corporal Harold Dawson. Your witness. Mr. McGuire, have you questioned Dawson about the fence line shooting? Yes. He claims to have been engaged in some manner by the enemy. - You don' believe him? - It's not my place-- Lance Corporal Dawson's been charged (..)1992 A Few Good Men
(..) we were taken under arrest. The night of August , did you fire a shot across the fence line into Cuba? - Yes, sir. - Why? - My mirror engaged, sir. - His mirror? Every American sentry post has a Cuban counterpart, called 'mirrors.' Lance Corporal's claiming that his mirror was about to fire at him. (..)1992 A Few Good Men