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Trial history biases in decision-making tasks are thought to reflect systematic updates of decision variables, therefore their precise nature informs conclusions about underlying heuristic strategies and learning processes. However, random drifts in decision variables can corrupt this inference by mimicking the signatures of systematic updates. Hence, identifying the trial-by-trial evolution of decision variables requires methods that can robustly account for such drifts. Recent studies (Lak'20, Mendonça'20) have made important advances in this direction, by proposing a convenient method to correct for the influence of slow drifts in decision criterion, a key decision variable. Here we apply this correction to a variety of updating scenarios, and evaluate its performance. We show that the correction fails for a wide range of commonly assumed systematic updating strategies, distorting one's inference away from the veridical strategies towards a narrow subset. To address these limitations, we propose a model-based approach for disambiguating systematic updates from random drifts, and demonstrate its success on real and synthetic datasets. We show that this approach accurately recovers the latent trajectory of drifts in decision criterion as well as the generative systematic updates from simulated data. Our results offer recommendations for methods to account for the interactions between history biases and slow drifts, and highlight the advantages of incorporating assumptions about the generative process directly into models of decision-making.