Conventional gradient damage models do not fully satisfy the convergence and monotonicity requirements of damage profiles, leading to spurious damage growth and artificially enforced irreversibility. Existing strategies that achieve convergence through empirical evolution of nonlocal interactions remain largely phenomenological, while irreversibility is typically ensured only in models tailored to strongly brittle materials. We address these fundamental issues through two key innovations. Firstly, an explicit solution for the final damage bandwidth is derived, which identifies the unbounded nonlocal variable as the cause of spurious damage growth in conventional models. To remedy this, an energy dissipation function, which vanishes the driving force with the increasing nonlocal variable, is proposed herein. This constrains the nonlocal variable to a finite interval [1,m] and thus ensures a convergent damage profile. Secondly, a general framework is developed to decouple the damage profile and softening behavior by introducing an additional dissipation term. This allows for a precise definition of damage evolution and softening behavior separately, preventing a pathological damage healing phenomenon. On this basis, by solving a Volterra integral equation of the first kind, arbitrary cohesive laws can be incorporated into the proposed model, while maintaining damage irreversibility. The model is validated by numerical examples with different traction-separation laws, which furthermore demonstrates its dual 𝛤-convergence and parameter-insensitive mechanical responses. The proposed approach provides a general and flexible framework to jointly describe the convergent and monotonic damage evolution process for arbitrary quasi-brittle materials using gradient damage models.