Context. Neutral sodium was the first atom that was detected in an exoplanetary atmosphere using the transmission spectroscopy technique. To date, it remains the most successfully detected species due to its strong doublet in the optical at 5890 Å and 5896 Å. However, the center-to-limb variation (CLV) of these lines in the host star can bias the Na I detection. When combined with the Rossiter-McLaughlin (RM) effect, the CLV can mimic or obscure a planetary absorption feature if it is not properly accounted for. Aims. This work aims to investigate the impact of three-dimensional (3D) radiation hydrodynamic stellar atmospheres and non-local thermodynamic equilibrium (NLTE) radiative transfer on the modeling of the CLV+RM effect in single-line transmission spectroscopy to improve the detection and characterization of exoplanet atmospheres. Methods. We produced a grid of 3D NLTE synthetic spectra for Na I for FGK-type dwarfs within the following parameter space: Teff = 4500–6500 K, log g = 4.0–5.0, and [Fe/H] = [‑0.5, 0, 0.5]. This grid was then interpolated to match the stellar parameters of four stars hosting well-known giant exoplanets, generating stellar spectra to correct for the CLV+RM effect in their transmission spectra. We used archival observations taken with the high-resolution ESPRESSO spectrograph. Results. Our work confirms the Na I detections in three systems, namely WASP-52b, WASP-76b, and WASP-127b, also improving the accuracy of the measured absorption depth. Furthermore, we find that 3D NLTE stellar models can explain the spectral features in the transmission spectra of HD 209458b without the need for any planetary absorption. In the grid of stellar synthetic spectra, we observe that the CLV effect is stronger for stars with low Teff and high log g. However, the combined effect of CLV and RM is highly dependent on the orbital geometry of the planet-star system. Conclusions. With the continuous improvement of instrumentation, it is crucial to use the most accurate stellar models available to correct for the CLV+RM effect in high-resolution transmission spectra to achieve the best possible characterization of exoplanet atmospheres. This will be fundamental in preparation for instruments such as ANDES at the Extremely Large Telescope to fully exploit its capabilities in the near future. We make our grid of 3D NLTE synthetic spectra for Na I publicly available.