Does a sister's refusal to shake hands with her sister and her saying, "I will bear your sin and my sin equally, and I must not shake her hand again," mean that she bears the full sin of severing kinship ties and absolves the other sister from this sin?

Maintaining kinship ties is obligatory, so strive to maintain your relationship with your sister. If you do what you can and she insists on her opinion, then you are not sinful. And remember the saying of the Prophet, peace and blessings be upon him: "If what you say is true, then you are as if you are feeding them hot ashes, and there will always be a supporter from Allah with you against them as long as you continue to do that."