The transitive use (mid-13c.) supplanted Middle English sench (compare drink/drench) which died out 14c. Related: Sank; sunk; sinking. Sinking fund is from 1724. Adjective phrase sink or swim is from 1660s. To sink without a trace is World War I military jargon, translating German spurlos versenkt.
mid-12c., dien, deighen, of sentient beings, "to cease to live," possibly from Old Danish døja or Old Norse deyja "to die, pass away," both from Proto-Germanic *dawjan (source also of Old Frisian deja "to kill," Old Saxon doian, Old High German touwen, Gothic diwans "mortal"), from PIE root *dheu- (3) "to pass away, die, become senseless" (source also of Old Irish dith "end, death," Old Church Slavonic daviti, Russian davit' "to choke, suffer").
It has been speculated that Old English had *diegan, from the same source, but it is not in any of the surviving texts and the preferred words were steorfan (see starve), sweltan (see swelter), wesan dead ("become dead"), also forðgan and other euphemisms.
Languages usually don't borrow words from abroad for central life experiences, but "die" words are an exception; they often are hidden or changed euphemistically out of superstitious dread. A Dutch euphemism translates as "to give the pipe to Maarten."
Regularly spelled dege through 15c., and still pronounced "dee" by some in Lancashire and Scotland. Of plants, "become devitalized, wither," late 14c.; in a general sense of "come to an end" from mid-13c. Meaning "be consumed with a great longing or yearning" (as in dying to go) is colloquial, from 1709. Used figuratively (of sounds, etc.) from 1580s; to die away "diminish gradually" is from 1670s. To die down "subside" is by 1834. Related: Died; dies.
To die out "become extinct" is from 1865. To die game "preserve a bold, resolute, and defiant spirit to the end" (especially of one facing the gallows) is from 1793. Phrase never say die "don't give up or in" is by 1822; the earliest contexts are in sailors' jargon.
"Never look so cloudy about it messmate," the latter continued in an unmoved tone—"Cheer up man, the rope is not twisted for your neck yet. Jack's alive; who's for a row? Never say die while there's a shot in the locker. Whup;" [Gerald Griffin, "Card Drawing," 1842]