Hi, Mia,
Sometimes iMovie preferences can get disrupted. Try deleting preferences to get a reset. It is a safe procedure that will not cause any loss of data or projects. To delete preferences open iMovie while holding down the Option and Command keys and select to delete preferences in the box that appears. iMovie will open in a new library. Reopen your old library to get back to your projects. Now see if you can drag in your music.
Also, you can try dragging in your music directly from the iTunes library into an iMovie project, or dragging to your desktop and from there into iMovie, and see if that works.
Songs that are in the cloud would need to be downloaded before dragging into iMovie.
-- Rich