The recently proposed mixed displacement-pressure-phase field formulation characterized by damage-mitigating incompressibility constraints excels in fracture modeling of nearly incompressible hyperelastic materials. However, such formulations generally call for mixed finite element (FE) allocations that satisfy the inf-sup condition, denying the popular lower equal-order linear FE pairs, like linear polynomial (P1) interpolation for all fields. To overcome this drawback, we develop a stabilized mixed formulation (P1/P1S) by resorting to a pressure projection technique. The lure of this formulation lies in its simplicity and versatility, allowing low-order P1 interpolation for all field variables. With this trait in mind, an efficient adaptive meshing strategy is designed, thereby speeding up computation by more than 15 times. For better coping with the adaptive scenarios involving crack nucleation, we also propose a novel energy-based mesh refinement criterion. The numerical treatment of the stabilized mixed FE formulation is elaborated, as well as the core operations of the adaptive meshing technique. The accuracy, efficiency, and robustness of the proposed formulation are validated by several representative numerical examples. Remarkably, a mixed I/II fracture test thereof is in good agreement with the experiment and thus promises to be a new benchmark.