As someone that had suffered from depression and anxiety related issues, I like to believe that that doesn't define my personality. Sometimes I still get affected by it, but with time I learned how to keep the whole thing in order, sometimes more successful than others, I admit. I don't believe that someone "chooses" or "not chooses" other person to a DD/lg relationship in regards of two different aspects of personality, since in my opinion, a DD/lg relationship develops with time, if she gets to know the part of you that's less happy (which is a possibility because those problems can be actually improved with care) she will also get to know the other parts of you, the good ones. I don't think there's a "you have this, you're out!" sort of thing, there are various types of littles , some will look at your depression and believe it's something that they don't want to mess with, there are others that will see you for who you are and will accept you, maybe see your whole depression/anxiety as a minor thing, or maybe as something that she might help you with.
Try not to forget that you're dealing with people, there are various factors to be taken to account, plus, a DD/lg relationship is still a relationship and like in every relationship, you have to find a partner that accepts you for who you are and nurture your qualities and accept that you're flawed person.
Have you looked for professional help? They might help when we're bouncing a little bit in life...
Good luck and here's some pizza!