Go back to your Warlock because after you got the Exotic Sword, the other swords should have been available for purchase from Lord Shaxx (Legendary Swords). Once you purchase the Void Edge or Sol Edge, then you can do the process like before to unlock the other Exotic's.
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So I have to obtain the other exotics using the same character? Or do I purchase the swords & give them to my other characters?
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If you got one exotic sword from a character, that character is able to purchase the legenary swords from Lord Shaxx. I used my Titan to get all 3 Exotic swords but its up to you if you want to use your other characters.