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No, he occupied Dol Guldur primarily as a location from which to search for the Ring
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Not at all. Sauron had no inkling what had become of the Ring until the very late Third Age. Circa TA 1000 he couldn't return to Mordor- Gondor was at the height of its imperial power, and Mordor was closely guarded. Why southern Mirkwood? The real reason of course was that T had said so in The Hobbit. But if we must back-write, it seems clear that Sauron wanted a place where he could remain concealed and his identity secret, while at the same time keeping an eye on his enemies. Amon Lanc was centrally located, hard to reach, and close to both the frontier provinces of Gondor and the main traffic artery of Middle-earth, as well as the Hithaeglir where Orcs still lurked. Lorien wasn't a factor, since Galadriel would not take up residence there for another 1800 years.
According to The Tale of Years, when Gandalf re-enters Dol Guldur in TA 2850, he learns who the Necromancer is, and that he is "seeking for news of the One." While Sauron would have known the Ring still existed, there is no reason at all to think that he knew what had happened to it, or even if it was still in Middle-earth. Only a century later, ca. 2939, does Sauron begin to search the River near the Gladden. From this Saruman concludes that Sauron "has learned of Isildur's end;" unsaid but just as important, Sauron has finally learned that Isildur took it in the first place.