What do you think of my idea of asking this girl out?

Don't ask her if she's single! That's a turn off! Just tell her you think she's super adorable and you should hang out. If she's not single then she'll say no because she has a boyfriend. It's definitely not embarrassing because it happens to everyone, but if she does turn you down, don't use that joke... it's weird. Just keep it smooth and be like "okay. let me know if you change your mind :)"