First of all, the stomach isn't really trying to "eat itself", although it feels like it. Also, your organs will not "shut down" or else you would die.
After your body realizes that making you miserable isn't working, it will begin to break down your muscle and fat stores for its protein and energy needs. At first your cells will use fatty acids and ketones as a source of energy and the liver will convert proteins and glycerol from fat into glucose for those cells that prefer it. After about a week most of the body, including the brain, will have shifted over to the use of ketone bodies as energy. Obviously the body goes into a much lower metabolic state to conserve energy, resulting in fewer bowel movements and an increased need for sleep/rest.
The liver will be hard at work attempting to create cholesterol, saturated fats and other nutrients that are normally derived from food. In fact many people experience very high cholesterol levels during starvation.
Upon reintroduction of food the body will begin to store as much fat as possible before it returns to its original metabolic state. This prepares it for the next time a starvation crisis arises.