Membrane peeling is a challenging procedure in retinal microsurgery, requiring careful manipulation of delicate tissues by using a micro-forceps and exerting very fine forces that are mostly imperceptible to the surgeon. Previously, we developed a micro-forceps with three integrated fiber Bragg grating (FBG) sensors to sense the lateral forces at the instrument's tip. However, importantly this architecture was insufficient to sense the tissue pulling forces along the forceps axis, which may be significant during membrane peeling. Our previous 3-DOF force sensing solutions developed for pick tools are not appropriate for forceps tools due to the motion and intrinsic forces that develop while opening/closing the forceps jaws. This paper presents a new design that adds another FBG attached to the forceps jaws to measure the axial loads. This involves not only the external tool-to-tissue interactions that we need to measure, but also the adverse effect of intrinsic actuation forces that arise due to the elastic deformation of jaws and friction. In this study, through experiments and finite element analyses, we model the intrinsic actuation force. We investigate the effect of the coefficient of friction and material type (stainless steel, titanium, nitinol) on this model. Then, the obtained model is used to separate the axial tool-to-tissue forces from the raw sensor measurements. Preliminary experiments and simulation results indicate that the developed linear model based on the actuation displacement is feasible to accurately predict the axial forces at the tool tip.