Fixed-item cognitive screening tools such as the Montreal Cognitive Assessment (MoCA) may ask redundant questions and increase patient burden. We propose AQS, an adaptive cognitive screening framework that selects questions and stops assessment according to an evolving cognitive state. Using ADNI-derived MoCA records, we construct an expanded question bank and an 8-dimensional state space, where free-form responses are mapped to discrete severity signals for online impairment updating. We compare handcrafted adaptive selection, fixed stopping, and reinforcement learning (RL) policies. Results show that combining a structured adaptive selector with an RL-based stopping policy yields the best performance–efficiency trade-off. Compared with the classical FisherInfo strategy, it improves Acc from 0.682 to 0.715 and AUC from 0.762 to 0.771, while reducing the average number of questions from 14.43 to 9.78. It also concentrates assessments in a more balanced question-count range and reduces excessive long-tail testing. Analysis further suggests that question selection and stopping are better treated as decoupled decisions, since naive joint RL leads to unstable optimization. These findings provide a simulation-grounded offline framework for efficient cognitive screening under data-scarce medical settings.