Well, bad news. Got a new disc this morning and it's doing the same thing. My only guess is that for some reason some of the older Xbox 360 models are being affected. Which is absolutely lovely, btw.
Unfortunately, you're problem is probably with your xbox itself... I've seen this issue before with other games. I bought Halo Reach when it first came out and had the same issue. wouldn't start or wouldn't get past the first autosave, gave disk unreadable, etc. older games worked fine, just the one game failed.
It happened to a number of other people, too and there was much complaining online. eventually people figured out that it was an issue with older 360 consoles. the error messages aren't real clear on the xbox (thanks Microsoft...), but in most cases when the error reads "disk unreadable" it's not talking about your game CD, it means the HDD.
I know its an expensive and annoying solution, but you may have to replace your console. I replaced mine and it solved all of the issues. I haven't gotten my copy of skyrim yet to see if it works (had to work early today so no midnight release for me...), but the issue sounds very similar to old issues with other games. if it works on another xbox console, that should confirm this....