Your title is misleading as OBS most likely has nothing to do with your issue
Sounds like your content (not the tool used to composite/send it) is the issue
There are lots of videos/tutorials on how to get properly licensed for certain music, especially in a House of Worship (HoW) setting, including discussion covering what some churches do that violates copyright (short answer - no, being a HoW doesn't exempt one from copyright law), and YouTube, FaceBook, and others will block any video which has such content
Fortunately playing pre-recorded popular music and livestreaming is not a challenge I've had to address (all of our music is played live, and older and largely public domain (ex. Hymnal) content)