If you are posting via YouTube, you can override the initial refusal notice, if the music is incidental to the video. You would have to go check how to do it, but I have done that in the past with one of my clips I created ages ago. I have currently been using mashups, which are musical interpretations based upon several music tracks for the audio backing of my learning to fly series of videos; but will move on to using royalty free music when I am finally at a stage of getting serious about the videos I release. At this point I am still learning to fly, capture good footage, & then edit that footage.
This was probably the holy grail of links that I found to background music options, both free & paid.
70+ Places to Find Background Music for Your Videos