The Collaborative International Dictionary of English v.0.48:

Drowse \Drowse\, v. t.
   To make heavy with sleepiness or imperfect sleep; to make
   dull or stupid. --Milton.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Drowse \Drowse\, n.
   A slight or imperfect sleep; a doze.
   [1913 Webster]

         But smiled on in a drowse of ecstasy.    --Mrs.
                                                  Browning.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Drowse \Drowse\ (drouz), v. i. [imp. & p. p. Drowsed (drouzd);
   p. pr. & vb. n. Drowsing.] [AS. dr[=u]sian, dr[=u]san, to
   sink, become slow or inactive; cf. OD. droosen to be sleepy,
   fall asleep, LG. dr[=u]sen, druusken, to slumber, fall down
   with a noise; prob, akin to AS. dre['o]san to fall. See
   Dreary.]
   To sleep imperfectly or unsoundly; to slumber; to be heavy
   with sleepiness; to doze. "He drowsed upon his couch."
   --South.
   [1913 Webster]

         In the pool drowsed the cattle up to their knees.
                                                  --Lowell.
   [1913 Webster]

WordNet (r) 3.0 (2006):

drowse
    n 1: a light fitful sleep [syn: doze, drowse]
    v 1: sleep lightly or for a short period of time [syn: snooze,
         drowse, doze]
    2: be on the verge of sleeping; "The students were drowsing in
       the 8 AM class"

Moby Thesaurus II by Grady Ward, 1.0:

33 Moby Thesaurus words for "drowse":
   beauty sleep, beddy-bye, bedtime, blanket drill, bye-bye, dorm,
   doze, dreamland, fitful sleep, hibernation, land of Nod,
   light sleep, nap, oversleep, repose, saw logs, saw wood, shut-eye,
   silken repose, sleep, sleep soundly, sleepland, sleepwalking,
   slumber, slumberland, snoozle, snore, somnambulism, somniloquy,
   somnus, take a nap, unconsciousness, winter sleep