The Shifted Boundary Method (SBM) is an alternative approach to traditional unfitted FEM approaches that addresses the small cut-cell problem by eliminating the need to perform cell cutting altogether. The SBM employs the use of surrogate domains, which contain only fully formed elements by design (no cuts). The true boundary conditions are shifted onto the location of the surrogate domain boundary by way of field extension operators constructed with Taylor expansions, preserving accuracy. The implementation of the Shifted Boundary Method is simple for Dirichlet boundary conditions. However, shifting Neumann boundary conditions required in the past a modified approach, leveraging a mixed formulation, since the higher-order terms in the Taylor expansion are unavailable (for example, all terms beyond first-order derivatives are zero for linear elements). We propose an alternative approach to Neumann boundary conditions that does not require the use of a mixed formulation, maintaining the same data structure requirements as the Dirichlet SBM boundary conditions. The new approach maintains optimal accuracy even when combined with a primal (irreducible) formulation of the variational equations. This result is achieved by an approximate integration formula of the governing equations in the gap between the true and surrogate boundary. Note that no cut-cell integration is performed, as the new approach involves only integrals on the surrogate boundary.