Drowsy

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late 15th century: probably from the stem of Old English drūsian ‘be languid or slow’, of Germanic origin; related to dreary.

wiktionary

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From drowse +‎ -y, despite the fact that drowsy (1520) is recorded before drowse (1570). Compare Old English drūsian(“to droop, drowse, become languid”).

etymonline

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drowsy (adj.)

"inclined to sleep, sleepy," 1520s, probably ultimately from Old English drusan, drusian "sink," also "become languid, slow, or inactive" (related to dreosan "to fall;" see dreary). There is no record of it in Middle English. Related: Drowsily; drowsiness.