Aha! Der OED meint, das sei AE, und alles andere als neu:
" b. A drawer. U.S.
1692 in Connecticut Probate Rec. (1904) I. 463, I giue to Elizabeth Thomson..one table with a draue in it.
1748 N.H. Probate Rec. III. 565, I give..my chist of draws to my dafter Lidea.
1764 Boston Evening Post 30 Jan. A Chest of Draws.
1775 in Essex Inst. Hist. Coll. (1877) XIII. 188 You know I can take a Draw at a time and lay them in the same manner into Dr Gardners.
1829 in W. L. Mackenzie Lives Butler & Hoyt (1845) 50 That celebrated receptacle of Chancery papers..the draw or bushel-basket..of his venerable predecessor.
1862 J. R. Lowell Biglow Papers 2nd Ser. iii. 108 Once git a smell o' musk into a draw An' it clings hold.
1898 E. N. Westcott David Harum xvi. 143 They're in the draw there.
1929 in H. Wentworth Amer. Dial. Dict. (1944) 178/2 The draw sticks.
1971 Amherst (Mass.) Record 28 July 15/1 Wanted to Buy. Two draw file and adding machine."