I am not familiar with c-dramas since I don't watch them, but maybe the music is just unreleased. That happens often with dramas of other countries.
I know for sure Hong Kong dramas like to use BGM from music production companies often. Here's one of the sites I used to look at to find some of their music. https://www.universalproductionmusic.com/en-us
Another method is uploading the song to Youtube (just pretend to make a video with some random picture and upload it) and the copyright info will show up after if it's copyrighted already. I found a few tracks from k-dramas like this before, thinking it was an unreleased track but they actually used music from non-asian music composers. From what I noticed so far is that Youtube can recognize more tracks not limited to Shazam or other song recognition apps.