Please accept my condolences in the loss of your husband. That issue in itself ranks in studies as one of the hardest life experiences. To have to deal with the
issues you mentioned makes it even more difficult.
The simple answer to your question is "no." You indicate that you didn't sign anything that indicated you were taking responsibility for the funeral expenses. Did you however, sign any document? If you did, was it entitled "Statement of Goods and Services?" That form lists the prices next to the appropriate services and merchandise that were selected. The law requires that at the time of the funeral arrangements, this document must be given to and signed by the person taking responsibility. Unless you happened to sign that document, you are not responsible.
Your situation does raise a couple of concerns however. If your husband was in South Carolina when he died, whether on business or vacation, it's very strange that you would not be the one to make all decisions regarding the funeral and care of his body. Although in another state at the time of his death, you are still his wife and therefore, his legal next of kin. I'm surprised that a funeral director would make funeral arrangements with his sister, unless he/she didn't know that he had a living wife. If you were separated but still legally married, the funeral arrangements would still be your responsibility unless you relinquished it with the funeral director's knowledge.
Please be sure to call the funeral director and explain your sister-in-law's request that you make payment and that you were not consulted or even involved with the arrangements SHE made. Ask if she or who else signed the form mentioned above. The funeral director should be able to give you some peace of mind with this. If not, please let me know.