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]
(..) Oh. Do you remember me? Yeah. I remember you too. You-You're exactly the same. Smiley and huggable and- - Ooh, and engaged? - What? Oh, yeah. - Holy shit. To who? - No one you know. Champagne. Now, come with me. (..)2005 In Her Shoes
(..) Oh, please! What we- Because of you, I have no idea where my sister is. I can't reach her. Her phone's cut off. She doesn't even know I'm engaged. My own sister. My best friend. And the worst part is, I can't talk to anyone about it. If I tell my father, he'll get (..)2005 In Her Shoes