Hey!
First off, I love how that sounds. I tried writing a book with a social media star MC recently, but couldn't get into it. I'm glad you can! I would read that!
So, I see your issue. As someone who recently finished a book with a lot of singing it it, I know that it can be kinda weird writing:
"Deeeeespaaaaaciiito, Quiero respirar tu cuello des-pa-ci-to, deja que te diga cosas al oíííííííííííído, para que te acuerdes si no estás conmigo. . ." she sang.
For your specific question, I suggest what you said about describing the emotions and body language, but not the thoughts unless they're watching it. What I mean is, when they're recording the songs, just (occasionally) go for their body language and a bit of emotion. Like:
"Deeeeespaaaaaciiito, Quiero respirar tu cuello des-pa-ci-to, deja que te diga cosas al oíííííííííííído, para que te acuerdes si no estás conmigo. . ." I threw my hair over my shoulder and closed my eyes, feeling the beat pound in my chest. "Despacito, quiero desnudarte a besos despacito, firmo en las paredes de tu laberinto, y hacer de tu cuerpo todo un manuscrito. . ."
It sounds better than:
"Deeeeespaaaaaciiito, Quiero respirar tu cuello des-pa-ci-to, deja que te diga cosas al oíííííííííííído, para que te acuerdes si no estás conmigo. . ." I sang. I wonder how I sound. "Despacito, quiero desnudarte a besos despacito, firmo en las paredes de tu laberinto, y hacer de tu cuerpo todo un manuscrito. . ." I straightened my posture and felt the happiness of the song.
Yaknow?
When they're watching the video, I think it'd be cool for them to be like:
"Despacito, quiero desnudarte a besos despacito, firmo en las paredes de tu laberinto, y hacer de tu cuerpo todo un manuscrito. . ." the recording sang. Ooh, I actually got that high note, I chuckle. Pulling me knees to chest, I continue watching.
I would suggest keeping emotions and body language in when they're recording, but not often; wait a bit between uses.
NEXT!
Yeah, the idea of a song list is awesome! Write the name of the song, the artist, what character sang it and on what page. WAY more specific.
If you wanted to, you could also use footnotes. The book I'm currently writing has some Japanese words in it, and I've found that it's easy and fast to type your word, click 'insert footnote', and type the definition at the bottom of the page.
If I were you, I think I would do the first idea, but whatever floats your boat!
Good luck!