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]
(..) This is for your engagement, man. You're a lucky, lucky dude. And, uh, congratulations to you. Oh, right. No, no, no. That's not happening any more. We're not, uh- We're not engaged. You're not getting married? Nah. Oh! You're Charlie Kelly, right? Yeah, yeah, yeah, yeah. (..)2009 TvShow: It's Always Sunny in Philadelphia Title: The Waitress Is Getting Married Season: 5 Episode 5
(..) This is because I was. But that-that didn't work out. Why are you asking? Are you single? What's your situation? Um, I'm engaged. But we're having a whole nontraditional kinda small thing. So you should meet her. Babe. Well, nah... - Yeah? (..)2009 TvShow: It's Always Sunny in Philadelphia Title: The Waitress Is Getting Married Season: 5 Episode 5