Heat oven to 350°F. Coat 13 x 9 x 2-inch baking pan with nonstick cooking spray.
In a small bowl, whisk together flour, baking soda, cinnamon and salt. Coarsely chop apples (with skins on) in a food processor. Set aside.
In a large bowl, with an electric mixer, cream together both kinds of sugar, shortening and eggs until smooth, about 2 minutes. Stir in flour mixture, chopped apples and nuts. Scrape into prepared pan and smooth top. Bake at 350°F for 50 to 60 minutes or until dark brown and firm in the center when pressed with fingertips. Cool in pan on wire rack.
In a small saucepan, combine both kinds of sugar, the butter, cream and corn syrup. Bring to a boil over high heat. Boil for 30 seconds. Remove from heat. Beat until cool, about 3 to 4 minutes. Stir in vanilla and rum flavoring. Cut cake into squares and drizzle with warm caramel. Note: If short on time, you may use jarred caramel sauce, heated slightly.