Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, 2 tablespoons granulated sugar, baking powder, and salt in a large bowl, stirring with a whisk. Cut in butter with a pastry blender or two knives until the mixture resembles coarse meal.
Combine buttermilk, oil, vanilla, and egg in a small bowl, stirring with a whisk. Add to flour mixture, stirring just until moist (dough will be sticky).
Turn dough out onto a baking sheet coated with cooking spray. Pat dough into an 8-inch circle. Cut dough into 8 wedges, cutting into, but not through, dough. Sprinkle 1 1/2 teaspoons turbinado sugar over dough. Bake at 425° for 15 minutes or until golden. Cool for 15 minutes.
Combine jelly, juice, and remaining 1 tablespoon granulated sugar in a microwave-safe bowl; microwave at high 1 minute. Stir mixture with a whisk until smooth. Combine jelly mixture, strawberries, and blackberries in a large bowl; toss to coat berries. Cover and chill.
Cut shortcake into 8 wedges. Cut each wedge in half horizontally; spoon 1/2 cup berry mixture over bottom half of each shortcake. Replace top half of shortcake. Top each serving with 2 tablespoons whipped topping; sprinkle each evenly with remaining 2 teaspoons turbinado sugar.