Hi there. Unfortunately there is currently no way to change a character's appearance. The only options are to continue with what you have or create a new character with the look you want.
Bungie has been aware of players' desire to change character appearances in-game. If you'd like them to prioritize it more, I would recommend you post your suggestions in the #Feedback forum to get their attention.
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Creating a new character would require rerunning the story, and reacquiring stasis correct?
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[quote]Creating a new character would require rerunning the story, and reacquiring stasis correct?[/quote] Correct.
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Yes, if you create a new character it has to go through the A Guardian Rises campaign and you would need to redo the Beyond Light campaign to unlock Stasis. You would also need to redo the A Mysterious Disturbance mission to unlock the Moon and Last Call for the Tangled Shore.