I have a daughter who is very like you describe yourself to be. I could never cut her off. Ever. Just because someone is more "sensitive" and anxious, does not invalidate any of those feelings. So please, do not be hard on yourself, relationships have 2 sides, and it is not always that one style is "right" and the other wrong. So NO , you are not "the problem".
But... Sometimes I do sigh when I see her number come up on my phone, or after a day or 2 of a visit.
so, perhaps I can show you the other side...
I sigh because we only get the bad times, the times she is feeling low. We would like some of the joy too.
We would also like for her to sometimes, just sometimes, stop talking and listen.
Can't tell her any bad news because it becomes part of her problems to deal with.
So I do identify a bit with your sister walking on eggshells.
But, the good news is that she is a loving, empathetic woman, and, she is now going to a therapist to talk through coping and "letting it go" strategies, which do seem to be helping, her mental health is becoming more robust.
She does now share something good from the present, without going straight on to worries and woes. And now, we also ask her about things, and feel more able to tell her our news, not scared to start off a 2 hour talking to.