Recent advances in neural radiation fields (NeRF) and 3D Gaussian-based SLAM have achieved impressive localization accuracy and high-quality dense mapping in static scenes. However, these methods remain challenged in dynamic environments, where moving objects violate the static-world assumption and introduce inconsistent observations that degrade both camera tracking and map reconstruction. This motivates two fundamental problems: robustly identifying dynamic objects and modeling them online. To address these limitations, we propose CAD-SLAM, a Consistency-Aware Dynamic SLAM framework with dynamic-static decoupled mapping. Our key insight is that dynamic objects inherently violate cross-view and cross-time scene consistency. We detect object motion by analyzing geometric and texture discrepancies between historical map renderings and real-world observations. Once a moving object is identified, we perform bidirectional dynamic object tracking (both backward and forward in time) to achieve complete sequence-wise dynamic recognition. Our consistency-aware dynamic detection model achieves category-agnostic, instantaneous dynamic identification, which effectively mitigates motion-induced interference during localization and mapping. In addition, we introduce a dynamic-static decoupled mapping strategy that employs a temporal Gaussian model for online incremental dynamic modeling. Experiments conducted on multiple dynamic datasets demonstrate the flexible and accurate dynamic segmentation capabilities of our method, along with the state-of-the-art performance in both localization and mapping.