The One Ring

As far as I know, Mirkwood remained dark until the Fourth Age. Encyclopedia of Arda says: "The stronghold of Sauron in the southern regions of Mirkwood, where he dwelt in secret as the Necromancer, until discovered by the Wise. When they assaulted him, he withdrew to Mordor, but the Nazgûl soon returned to reclaim Dol Guldur." Which I think would be no less dark 😉 And I do think Legolas said something about his home still being dark during the time of LotR.

Yes you are right, After Sauron was reduced to a powerless "spirit of malice" at the conclusion of the Third Age, the darkness was lifted from Mirkwood, and it became known again by its old name of Eryn Lasgalen, Sindarin for the Wood of Greenleaves.