Current research in online time series forecasting (OTSF) faces two significant issues. The first is information leakage, where models make predictions and are then evaluated on historical time steps that have already been used in backpropagation for parameter updates. The second is practicality: while forecasting in real-world applications typically emphasizes looking ahead and anticipating future uncertainties, prediction sequences in this setting include only one future step with the remaining being observed time points. This necessitates a redefinition of the OTSF setting, focusing on predicting unknown future steps and evaluating unobserved data points. Following this new setting, challenges arise in leveraging incomplete pairs of ground truth and predictions for backpropagation, as well as in generalizing accurate information without overfitting to noise from recent data streams. To address these challenges, we propose a novel dual-stream framework for online forecasting (DSOF): a slow stream that updates with complete data using experience replay, and a fast stream that adapts to recent data through temporal difference learning. This dual-stream approach updates a teacher-student model learned through a residual learning strategy, generating predictions in a coarse-to-fine manner. Extensive experiments demonstrate its improvement in forecasting performance in changing environments. Our code is publicly available at https://github.com/yyalau/iclr2025_dsof.