I had this problem too, my new would show the library name of the old but wouldn't actually show the music. What work for me is updating both computers to have the latest version of Itunes. For me, my Mac (new computer) had the latest Itunes and my PC (old computer) had a slightly older version. After I updated the Itunes on the PC I was able to transfer my music from old to new.
You can try that but I don't know if it will work like it did for me. Good luck!