Great work on beginning to think about how you can get better! Going out on your own and becoming more independent is a great way to work through anxiety! As for your sister, let of your anger towards her. She can make her own choices.
As for you, it is possible to live on your own if you work a job and have a studio or one bedroom apartment, unless of course the only jobs you can get are really crummy ones. If they are, then begin to work your way towards independence. Work a crummy job now so that you can save up some money, get a better job, and eventually move out. Or, go to college and get the skills you need to get the job you think will make you happy. The choice is yours.
It is not necessary to move to another state-you could move an hour or two away, which is enough to keep your family from visiting you too often, but they are close enough to help you in time of emergency. You can do it, and this is scary for anyone, but figure out a plan that works for you and go with it.
Just be sure to create a goal, in this case independence, and figure out a way to get to it, because a way always exists. Good luck and be sure to ask if you have questions.