Sentence:  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 <WSD>drowsed</WSD> his way through the first half hour of the treatment, ... 
Meaning of the word marked with <WSD> tags:  To advance drowsily. (Used especially in the phrase "drowse one's way" ⇒ sleepily make one's way.)