Advection-dominated convection--diffusion and convection--diffusion--reaction equations in three spatial dimensions present persistent numerical challenges, particularly in regions characterized by sharp boundary layers, internal shocks, and steep solution gradients. Classical stabilized finite element methods, such as the streamline-upwind/Petrov--Galerkin (SUPG) formulation augmented with discontinuity-capturing operators, effectively suppress spurious oscillations inherent in standard Galerkin (GFEM) approximations. However, these advanced stabilization mechanisms may introduce excessive numerical dissipation that smears sharp fronts and compromises solution accuracy in critical regions. This work presents a novel two-stage hybrid computational framework that synergistically combines the robustness of stabilized finite element methods with the approximation capabilities of Physics-Informed Neural Networks (PINNs) for solving three-dimensional transport problems. In the first stage, the SUPG formulation enhanced with a YZ$\beta$ discontinuity-capturing operator is employed to obtain numerically stable preliminary solutions. While this approach successfully eliminates non-physical oscillations, it may introduce artificial diffusion, particularly in regions featuring steep gradients. In the second stage, a PINN-based correction mechanism is deployed to identify and mitigate the excessive dissipation introduced by the stabilization procedure. The neural network architecture incorporates Fourier feature embeddings for enhanced high-frequency representation and deep residual blocks for improved gradient flow. A multi-phase adaptive weight scheduling scheme is implemented to gradually transition from data-dominant to physics-dominant training, ensuring stable convergence while maintaining physical consistency. Numerical experiments on benchmark three-dimensional advection-dominated problems demonstrate that the proposed hybrid framework achieves superior accuracy in boundary layer regions compared to standalone SUPG-YZ$\beta$ solutions. The methodology successfully balances the trade-off between numerical stability and solution accuracy, producing oscillation-free results with sharply resolved gradients. All finite element computations are performed using the FEniCS platform, while PINN training is conducted in PyTorch with full GPU acceleration.