When I got a new computer, I faced the same problem. Ultimately I had to redo all my lists, but I redid them in iTunes, and they are part of my iTunes XML file and database file now, so the next time I switch computers, theoretically, they will stay intact within those files. I keep multiple backups just in case.

Other than that, the logical answer to your situation now is, "Whatever software you used to store your playlists on your old computer, copy that to your new computer." Of course it sounds like you tried that and it didn't work.

I hope someone else can offer more help than I.