A more accurate analytical solution of the vertical-current approximation nonlinear force-free field (VCA3-NLFFF) model is presented that includes, besides the radial (Br) and azimuthal (B{phi}) magnetic field components, a poloidal component (B_{theta}_/=0) as well. This new analytical solution is of second-order accuracy in the divergence-freeness condition and of third-order accuracy in the force-freeness condition. We reanalyze the sample of 173 GOES M- and X-class flares observed with the Atmospheric Imaging Assembly and Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory (SDO). The new code reproduces helically twisted loops with a low winding number below the kink instability consistently, avoiding unstable, highly twisted structures of the Gold-Hoyle flux rope type. The magnetic energies agree within E_VCA3_/E_W_=0.99{+/-}0.21 with the Wiegelmann (W-NLFFF) code. The time evolution of the magnetic field reveals multiple, intermittent energy buildup and releases in most flares, contradicting both the Rosner-Vaiana model (with gradual energy storage in the corona) and the principle of timescale separation ({tau}flare<<{tau}storage) postulated in self-organized criticality models. The mean dissipated flare energy is found to amount to 7%{+/-}3% of the potential energy, or 60%{+/-}26% of the free energy, a result that can be used for predicting flare magnitudes based on the potential field of active regions.