How do you write a song fic with avoiding copy-write issues?

Well, what I do is add the lyrics, but it's more like they are listening to it in a club or on the radio. like example, for one of my stories the couple was in the club when the song came on but the premise of the song is in the story

Songfics aren't something I've written myself, but maybe if you limited the amount of lyrics you quoted that might be safer? Also it'd be a good idea to include the song in a disclaimer, saying that you don't own the song as well as the original fandom itself and saying who does own the song.

I'm pretty sure that stories which break the rules only get removed if reported - I think fanfiction.net is too large and involves too many works for their moderators to check everything on it themselves. So maybe the songfics by others you've seen weren't reported, and maybe your own songfic would be safer in a fandom which is generally OK with this genre of writing.