Lethargically tells you how someone does a thing: slowly, heavily, with little energy and little interest. The person is not just quiet; the action itself looks dragged. She moved lethargically around the kitchen. He answered lethargically, as if each word were extra work. It is the adverb of lethargic, and it names manner, not a feeling you “are” — you are lethargic; you walk lethargically. You will meet it more in writing than in quick talk. Same idea as the adjective: this is low energy, not a moral note about laziness.