So my used GOTY edition of Oblivion on Xbox is screwed up and freezes my Xbox everytime I go to load up a save. So when I buy a new version of Oblivion will the Shivering Isles and Knights Of The Nine work on the new version of Oblivion?

Id try starting a new game. If that doesn't work check if the disc is scratched.

As W2drs21 said, check and see if your disc is scratched. If so, there are ways to fix that. But, to answer your original question, yes your old saves will work with a new copy. At least, they will as long as they aren't corrupt.

Have you tried downloading the disc(s) to your Xbox 360? It might help the problem with your crashes and freezes.

I'm going to try doing that and hopefully i'll get it to work.

And to the other people there are many scratches on my disc. None of my saves say "corrupted" so i'm assuming that they are not and this is just a problem with the actual disc.