I have played Skyrim installed to my HDD and from the disc, yet I still get horrible textures.
However, I made two interesting discoveries.
Whenever I would load a game save, at first, everything looked more or less okay. But when I went through any menus by pressing B, everything got worse, especially my character's hands.
Also, a friend of mine who I'm talking to ATM has said he's got Skyrim installed on his HDD and is having no low textures whatsoever. I probed deeper into this mystery and discovered something. He has an older XB360 model (original Pro model) and I have a newer XB360 model (Slim Kinect ready model).
It seems to me that this issue depends on which XB360 model you are using.

)... I'm using an arcade from around '09 I think.
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