Take it with a grain of salt and those that speak very well, please correct my mistakes, but isn't "She said, "I am not fat." would be something like :
彼女は"私は太っていない。"と言います。
I'm not sure exactly on my は's and が's in this instance, but I know you need to state the person who is your topic and subject, and then end with と言う since that is what is being done by the topic/subject. At least every example I've seen has been this way.