I can relate to what you are living. It isn't just about your birthday, it is about everyday. The birthday is just what put you over the edge. It happened to me too. In fact, I found out after 40 years that it was my sister who caused it all. She literally lied cheated stole and manipulated to turn my parents against me.
I left town over it as well, because I knew that I didn't want to be exposed to others who did not have my best interest at heart. If they are forgetting your birthday then what else are they doing behind your back that forgetting your birthday was a simple sign of their disrespect, and how much is going on that made them decide to celebrate your sisters life and not yours. You have to consider these things. I learned the hard way that my family doesn't deserve me. They left all my baby pictures at a rental house when I was three because there just wasn't enough room in the moving truck for them, they bought my sister a car at 16, and my sister in law a car at 18, and not only did I not get a car, but no one would even take me to the DMV to take my driving test, or even loan me a vehicle to take it in. Think about things and see if you aren't being done wrong on different levels, and they always have some lame excuse to cover themselves.
I did leave town, for six years then realized that they weren't going to push me away from the city I loved. I moved back with my two young kids, which no one gets to see, their own bad, and when they want to see me i blow them off. I don't need it, and neither do you. There will be people out in the world, away from your selfish family who will be your new family, and who chooses to show you affection and care, and you won't have to ask for it. I have seen the world away from my family and it is real and there are people who are kinder and gentler and who care that others have feelings.
Go find it for yourself if you can take care of yourself financially. Good Luck and Happy Birthday.