Gollum never passed through Lothlorien. All he could do was skirt around its edges, presumably, because the "Elvishness" of the land was hurtful to him (the hithlain burned him later).
I think that Galadriel (and Celeborn) probably didn't discourage those rumors of "a nameless fear/unease" about Lothlorien because they would get less intruders and ill-wishers. Galadriel wanted to seal off Lothlorien (for the most part) from the harmful influences of the rest of Middle-earth, and a constant influx and exit of people wouldn't have enabled her to hold to that purpose.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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