This paper presents Divergence-Regularized Discounted Aggregation (DRDA), a multi-round learning system for solving partially observable stochastic games (POSGs). DRDA is based on action values and applicable to multiplayer POSGs, which can unify normal-form games (NFGs), extensive-form games (EFGs) with perfect recall, and Markov games (MGs). In each single round, DRDA can be viewed as a discounted variant of Follow the Regularized Leader (FTRL) under a general value function for POSGs. While previous studies on discounted FTRL have demonstrated its last-iterate convergence towards quantal response equilibrium (QRE) in NFGs, this paper extends the theoretical results to POSGs under divergence regularization and generalizes the QRE concept of Nash distribution. The linear last-iterate convergence of single-round DRDA to its rest point is proved under the assumption on the hypomonotonicity of the game. When the rest point is unique, it induces the unique Nash distribution defined in the POSG, which has a bounded deviation from Nash equilibrium (NE). Under multiple learning rounds, DRDA keeps replacing the base policy for divergence regularization with the policy at the rest point in the previous round. It is further proved that the limit point of multi-round DRDA must be an exact NE (rather than a QRE). In experiments, discrete-time DRDA can converge to NE at a near-exponential rate in (multiplayer) NFGs and outperform the existing baselines for EFGs, MGs, and typical POSGs.