No. In fact, if you unlock the exotic quest on a character that already has it active for another sword you can actually skip the Crucible step. Assuming you've already done the Crucible step for that first sword of course.
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okay thanks! I was freaking out until your reply. I have the darkdrinker and want all 3.
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No worries. I deliberately did all three exotic quests on one character, void and arc even at the same time. If you'd mess something up by transferring your sword between characters I'd have known. :P