I think the problem was not that Gawyn wasn't used to living in someone's shadow, it's that he wasn't happy about it. He always had hope for the future, that it would lead to glories that would allow him to step out from the shadows of his sister and brother, or whomever else. Instead, he gets exiled to a small town outside Tar Valon working for no good reason for people he hates while other people, especially the guy his sister likes that he thinks killed his mother, go about getting glory. This is what had made him so insufferable and twisted for so long, and it's something he didn't even recognize in himself.