(intransitive) To be sleepy and inactive also figurative.
1902, Jack London, Moon-Face:Under the aching noonday glare, when the green things drooped and the birds withdrew to the depths of the forest, and all nature drowsed, his great "Ha! ha!" and "Ho! ho!" rose up to the sky and challenged the sun.
1973 July, Melville Bell Grosvenor, Homeward with Ulysses, published in National Geographic, volume 144, number 1:In August the cicadas chorused, and the dusty olive trees drowsed in the sun.
(transitive) To advance drowsily. Used especially in the phrase "drowse one's way" ⇒ sleepily make one's way.
1873, Mark Twain, The Gilded Age: A Tale of Today (1915 republication), page 285:... the wary tadpole returned from exile, the bullfrog resumed his ancient song, the tranquil turtle sunned his back upon bank and log and drowsed his grateful life away as in the old sweet days of yore.
1966, John Cunyus Hodges, William Congreve, the man: a biography from new sources, page 25:Congreve held fast to the Greek poets, but otherwise seems to have drowsed his way through Trinity studies.
2002, Marsha Ward, The Man from Shenandoah, page 55Ida had kept him awake while he drowsed his way up the old King's Trace in eastern Missouri, feverish and weak.
2008, Sarah Mayberry, Cruise Control, published in Best of Makeovers Bundle, page 209:They were led into a large, attractive room with twin massage beds, and welcomed by their masseurs—in Balinese tradition, he had a male masseur, Anna a female. He drowsed his way through the first half hour of the treatment, ...
(transitive) To make heavy with sleepiness or imperfect sleep; to make dull or stupid.