Yeah they have moving boxes that fit records. I’d probably make stacks and package wrap them so they can’t shift. Then air pack on top. The boxes themselves are sturdy. Just reinforce the bottom with tape in both directions
Media Mail, well padded in a way to try to avoid damage. I think bubble wrap and combining CDs in ziploc bags would help.
Also if you have a cd book, you can put the discs and art in the book then just buy jewel cases off of amazon when you get home.