Skyrim disc damaged, if I buy another will I still be able to play my saved games?

Solution 1:

The optical discs used for games are a read only media, nothing you do in game is stored on them, only on the storage in your console.

As long as you buy a disc from the same region your original disc was, your progress and DLC should still work.

Solution 2:

Theoretically, it would be possible to press an unique readable serial number to each optical disk. When one would do that, the application could tell when it runs from a different physical disk. But in practice nobody does it, because it would make the manufacturing process a lot more expensive. So no, every Skyrim disk looks the same for your console.

By the way: some companies will send you a new disk when you send in a broken one. Remember, according to the game companies, you buy the right to play the game, not the ownership of the media it is on.