After a few setbacks a few years ago, I was able to fine-tune the quality of my shipping packaging. You can sleep well. This is totally safe.
A first packaging in bubble wrap (yes I know, it’s plastic but for the moment, nothing better). Each piece is then protected by cardboard (individual pack box for large pieces or separate cardboard for small). All of it goes in a box that goes in another box. That’s the best way to resist shocks, because I know how it is in transportation. The day I saw a documentary about the carriers, I had a bad night.
Just feel comfortable !