Edged weapons held no monopoly on bringing about the demise of Shakespeare's most ill-fated characters. See Weapons of Masterpuppet Destruction I. Poison is featured prominently in many of his melodramatic death scenes. Generally administered by scheming villains and villainesses, occasionally by thwarted lovers at their own hands, there's enough deadly toxin flowing throughout the plays that some of the fatal poisonings are accidental, like Hamlet's mother Gertrude. And to be fair, a number of those pointy daggers and rapiers are treated with nasty concoctions to guarantee efficacy. Just ask Laertes.
Nor do bottled potions constitute the only means of death by poison. Romeo consults an apothecary for his tincture, but Cleopatra goes right to the source and employs a handy asp to do herself in. Household items also serve as murder weapons. Unjustly accused of adultery, Desdemona is smothered by Othello with a pillow.