wad (n.)
early 15c., wadde, "small bunch of fibrous, soft material for padding or stuffing," of uncertain origin, perhaps from Medieval Latin wadda (14c., source also of French ouate, Italian ovate), or Dutch watten (source of German Watte), or Middle English wadmal (c.1300) "coarse woolen cloth," which seems to be from Old Norse vaðmal "a woolen fabric of Scandinavia," probably from vað "cloth" + mal "measure."

The meaning "something bundled up tightly" (especially paper currency) is from 1778. To shoot (one's) wad "do all one can do" is recorded from 1914. The immediate source of the expression probably is the sense of "disk of cloth used to hold powder and shot in place in a gun." Wad in slang sense of "a load of semen" is attested from 1920s, and the expression now often is felt in this sense. As a suffix, -wad in 1980s joined -bag, -ball, -head in combinations meaning "disgusting or unpleasant person."
wad (v.)
1570s, "put a wad into," from wad (n.). From 1670s as "form into a wad;" 1759 as "pad or stuff with wadding." Related: Wadded; wadding.