High-resolution uncertainty quantification requires the efficient solution of large-scale and high-dimensional stochastic partial differential equations. Existing stochastic spectral finite element methods combined with domain decomposition usually rely on solving augmented deterministic systems, whose size grows rapidly with the stochastic dimension, limiting their applicability to large-scale problems. In this study, we propose a new stochastic finite element framework that decouples the stochastic solution into deterministic and stochastic components, thereby avoiding augmented systems. Specifically, the stochastic solution is represented as a sum of products of deterministic vectors and random variables. The deterministic vectors are computed from deterministic finite element equations using a two-level additive Schwarz-preconditioned domain decomposition method, with coarse spaces constructed by Constant and GenEO (Generalized Eigenvalues in the Overlap) approaches, while the random variables are obtained from one-dimensional stochastic algebraic equations. The size of the resulting deterministic systems is comparable to that of the original physical problem, enabling efficient parallel scalability. Numerical experiments on two- and three-dimensional Poisson and elasticity problems with over one million degrees of freedom and up to one hundred random variables demonstrate the robustness and scalability of the proposed framework.