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Topology and shape optimization (TSO) is a powerful technique for achieving high-stiffness configurations of stiffened curved shells. However, it presents a challenge to obtain stiffener layouts that satisfy manufacturing constraints using topology optimization (TO) while also considering the changes in the curved shell shape through shape optimization (SO). There are three main contributions in this paper: (1) A shape-dependent TO method is developed for the design of stiffener layouts on different curved surfaces. (2) An integrated TSO framework is established by combining SO design variables and TO stiffener constraints through a mesh deformation approach. (3) A surrogate model of SO variables to TO result is constructed by a deep neural network (DNN) to accelerate the proposed framework. First, an actual optimization model including shape modeling and shape-dependent TO is established. The shape-dependent TO method is based on anisotropic filters to achieve stiffener constraints. Based on shape variables, mesh deformation is used to move the nodes to achieve the shape modeling without changing the mesh topology, and also to move the anisotropic filter direction to ensure the stiffener constraints. Then, sample points are generated in the shape design space as the input, and the TO results are obtained by the actual optimization model as the output. A DNN surrogate model is constructed and its optimal point is found. Finally, the DNN surrogate model is updated by filling in the optimal points to achieve the integrated TSO. Three engineering examples are carried out to illustrate the effectiveness of the proposed framework, including a spherical shell subjected to external pressure, a conical shell subjected to concentrated forces, and an S-shaped variable cross-sectional curved shell subjected to internal pressure. The results verify the effectiveness and outstanding optimization ability of the integrated TSO framework compared to the conventional methods such as single TO without shape change.