Our responses to ppl, all ppl, teach them how to treat us. If you continually give in, where is the insentive for your sister to stop begging when it's worked so well in the past. I can't make this decision for you as to what to do, to see her, to not see her. My advice is this, if she is that unhealthy for you to be around and brings you down and tears you down. Then you need to set a consistant and solid boundry that you will not see her until she gets some help. Provide her with names and numbers of places she can get help. Let the rest of the family know where you stand and these other ppl she enlists to call you and tell you to see her. They are probably just trying to pawn her off on you anyway, so they can get a reprieve.
If you feel strong enough to be around her and not succumb to any abuse, mistreatment or ill affect...set boundaries within the context your willing to see her. Only for so much time, at a determined place, with so and so present...these are just suggestions. I don't know the specifics of your situation. that being said, all I can do is reiterate...we teach people how to treat us. This is a good thing because then we have to power to change whatever might be wrong or harmful or abusive to us.
Good Luck. We're here for you. I'm glad you made your way to PC. Look forward to getting to know you better. TgrsPurr.