Yes, but you can hardly blame her. In her opinion, that's what defines her. She's known she was the avatar almost her whole life, it's no surprise she doesn't know who she is without that title.

She could bend air still but continued to mope when she lost the other elements. At least she could still bend something!

^Indeed.

I think it's a great flaw in her character to work with- hopefully they'll do that with the next season, even though it could have fit much better with season 1 with the whole "oppressing bender, non-benders have a disadvantage" thing. She could have learned to fight at the levels of non-benders, to make it more "equal." But I'm, again, nitpicking.

I think she can actually get unnecessarily violent when she's bending as well, which is another flaw to work with. ..I mean, sometimes when I see her fight I swear she can be 2 inches from killing somebody.

She is way to cocky and thinks she can get everything she wants since she is the Avatar. Then when she lost her bending, still having air, she cried in front of NON bender. Then when she could actually learn how to fight with just air, everything was just handed to her without learning anything.

No I don't think so :)