The long-standing paradox of the linear polarization signal of the Na I D1 line was recently resolved by accounting for the atom's hyperfine structure and the detailed spectral structure of the incident radiation field. That modeling relied on the simplifying angle-averaged (AA) approximation for partial frequency redistribution (PRD) in scattering, which potentially neglects important angle-frequency couplings. This work aims at evaluating the suitability of a PRD-AA modeling for the D1 and D2 lines through comparisons with general angle-dependent (AD) PRD calculations in both the absence and presence of magnetic fields. We solved the radiative transfer problem for polarized radiation in a 1D semiempirical atmospheric model with microturbulent and isotropic magnetic fields, accounting for PRD effects and comparing PRD-AA and PRD-AD modelings. The D1 and D2 lines are modeled separately as a two-level atomic system with hyperfine structure. The numerical results confirm that a spectrally structured radiation field induces linear polarization in the D1 line. However, the PRD-AA approximation greatly impacts the Q/I shape, producing an antisymmetric pattern instead of the more symmetric PRD-AD one while presenting a similar sensitivity to magnetic fields between 10 and 200 G. Under the PRD-AA approximation, the Q/I profile of the D2 line presents an artificial dip in its core, which is not found for the PRD-AD case. We conclude that accounting for PRD-AD effects is essential to suitably model the scattering polarization of the Na I D lines. These results bring us closer to exploiting the full diagnostic potential of these lines for the elusive chromospheric magnetic fields.