First, before I say anything, I just want to say that you're doing a great job and I'm really happy for you!
I think you're doing the right thing. I don't think your ex (or anyone) would/should mind if you don't attend for birthday gathering even if you're not busy, because after all you're trying to move on, and the best way to do it is stay clear from the ex. It's a normal thing, and it's very understandable, so don't think about it too much or worry about it (because there's nothing to worry about).
Stop caring how she would think about all of your actions, it doesn't matter. Focus on yourself. And from a woman's perspective (and from what you wrote on the last chapter post), I think you already left a great last impression on her, just let it stay that way! ( do you really want to hit yourself on the head later for ruining that image. From personal experience, I feel really embarrassed about how I acted after my ex broke up with me, and I feel bad that he probably think I'm some scary needy girl. It's probably a image that I can never erase (even though I wish I could get rid of it), but oh well what can I do.)
Also remember what you told her when you exchange thing? I can't exactly remember what you wrote, but I believe its something along the line of - " don't expect me to stay in touch." Why don't you do what you said? Ignore her text! I think if you stop replying to her, she would stop contacting you, and in turn it would make the process of healing yourself easier. I'm not saying that this is something easy to do, but it's probably will let you move on faster.
Well anyway this is my opinion and I wish you the best of luck!
Also I hope that you will find a career that you will love with a passion in the future! I think it will make you happier and more confident!