Wettability quantifies the affinity between a solid surface and a contacting fluid. This property governs a wide variety of biological, biomedical and industrial processes. The Lattice Boltzmann Method (LBM), combined with the Shan-Chen (SC) pseudopotential scheme, is a viable approach to investigate such adhesion mechanisms. The LBM is a mesoscopic Navier-Stokes solver based on the collision and streaming of fictitious particles on a lattice, while the SC scheme models multiphase flows by embedding a cohesive (fluid-fluid) non-ideal interparticle force, leading to a spontaneous phase separation below a critical temperature. To damp the formation of intense spurious currents, developing at fluid interfaces and leading to instability, and to increase the accuracy of the naive SC-LBM model, several features have been progressively implemented, such as a central moment-based (cascaded) collision operator, an entropic stabilization procedure, and a high isotropy order interaction stencil. The SC-LBM model has also been extended to investigate adhesive phenomena: several formulations of the fluid-solid interaction problem are presented in literature. Nevertheless, these still lack an effective integration with cohesive forces and fail to manage complex solid geometries. To address these issues, we developed a unified non-ideal force framework and implemented a mass-conserving Unfitted Method termed the Coupled Boundary Wettability (CBW) scheme. The Unidirectional CBW (U-CBW) formulation exploits local body force data to correct the mass leakage induced by boundary interpolations. Furthermore, its Bidirectional extension (B-CBW) allows to finely tune adhesion forces at solid boundary nodes by incorporating their off-grid positions. Extensive testing of both U- and B-CBW schemes within a cascaded and entropic SC-LBM framework confirms their stable and accurate behavior. Given the capabilities of the model, we consider the outcome of our work a relevant step towards a more accurate description of natural phenomena and engineering devices.