Hey there. In addition to the answer given, we've been given three name changes in total so you should be able to change again, if you've only used one.
If you can enter a new name into the name box, this is confirmation that you do have name changes available. If the name does not get accepted, ignore the 'you can only change your name once' message, it's misleading, the name is just hitting the filter and can't be used. Keep trying something else and you'll get one to go through eventually.
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And I'm assuming that if I have no changes left, my name isn't fixable until Bungie gives us another name change. Correct?
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That is correct yeah