To go along with what Zigur brings up with how Sauron originally was drawn Melkor's rebellion because he was you could say, star struck, but Melkor's splendor and ability to implement his designs relatively quickly.
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He [Sauron] still had the relics of positive purposes, that descended from the good of the nature in which he began: it had been his virtue (and therefore also the cause of his fall, and of his relapse) that he loved order and co- ordination, and disliked all confusion and wasteful friction. (It was the apparent will and power of Melkor to effect his designs quickly and masterfully that had first attracted Sauron to him.)~Home X: Morgoth's Ring; Myths Transformed
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Later, Morgoth spun downwards into nihilism, and Sauron never reached this stage of chaos, since he still had, what Tolkien calls "relics of positive purposes." I think, given enough time Sauron was following down the same path and would have become a nihilist:
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In all the deeds of Melkor the Morgoth upon Arda, in his vast works and in the deceits of his cunning, Sauron had a part, and was only less evil than his master in that for long he served another and not himself. But in after years he rose like a shadow of Morgoth and a ghost of his malice, and walked behind him on the same ruinous path down into the Void.~The Silmarillion: The Valaquenta
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I like how you break it down between Chaotic Evil and Lawful Evil. It sounds like two apt descriptions.