Peridynamics is a nonlocal continuum mechanics formulation that is particularly well-suited for simulating dynamic fracture in materials. The correspondence formulation addresses the limitations of earlier peridynamic formulations by enabling accurate elastic behavior and facilitating the use of classical constitutive models. However, it suffers from stability issues due to zero-energy modes, similar to those observed in other meshfree methods like the Reproducing Kernel Particle Method with nodal integration. Bond-associated peridynamics has been proposed as a stabilization technique that approximates the deformation gradient on a per-bond basis, similar to higher-order integration schemes in other meshfree methods. Regarding dynamic fracture simulations, the bond-associated approach significantly improves the stability. However, the traditional bond deletion schemes in peridynamics for modeling fracture lead to a new instability near cracks. When bonds are removed, the shape tensor becomes ill-conditioned, amplifying zero-energy modes and causing wrong solutions and instabilities. Proper regularization of the inversion of the shape tensor can be used to slightly improve this issue, but the overall problem lies in the bond-deletion schemes. This work introduces a novel phase-field inspired damage formulation that eliminates these pathologies through continuous stress degradation rather than bond deletion. The proposed model employs a quadratic degradation function applied to bond-level stresses, where the damage variable evolves based on a thermodynamically consistent crack driving force. This approach smoothly reduces bond contributions to the internal force while maintaining stable kinematics throughout the fracture process. An irreversible history variable ensures physically meaningful damage evolution without spurious healing. The formulation is implemented within the reproducing kernel correspondence framework using bond-associated quadrature points. Each bond tracks its own damage state based on either the maximum principal stress or tensile strain energy at the bond midpoint. The critical stress establishes a direct connection between material properties and the peridynamic horizon, ensuring mesh objectivity and convergence to Griffith fracture in the limit of vanishing regularization length. Numerical examples demonstrate the effectiveness of the proposed model in simulating dynamic fracture processes.