Is there a cure for stage 3 rectal cancer that doesn't affect continence?

Location dependent. The closer rectal cancer is to the anus, the more likely it can affect muscle control of continence. Most surgeons will offer chemotherapy and radiation before surgery for stage 3 rectal cancer. After the tumor shrinks, surgery can be easier in terms of location from the anal sphincter muscles. Incontinence is not an expected result in most cases.
YES, curative intent. For stage 3 rectal cancer- the goal of therapy is to cure the disease. Treatment will include- combination of chemotherapy and radiation therapy that is given prior to surgery;then followed by curative surgery-then followed by consolidation of chemotherapy. Discuss further with your oncology team (you would need to see a medical oncologist, radiation oncologist and surgical oncologist).Good luck!