I don't want to pay my sister a full month's rent if I left after a week
Why would you not pay her for staying?
I don't feel like I owe her anything.
If you agreed to pay for your share of the rent then you should do it. Not paying your sister could get your sister in financial trouble.
She doesn't need it. She can just get her own apartment.
It doesn't matter if you believe she doesn't need it. An agreement between siblings should be honored. It's bad to let money come between family members.