Floating wind turbines (FWTs) are key equipment for deep-sea clean energy exploitation, and their structural health condition is directly related to operational safety and energy output. However, FWT vibration signals exhibit significant non-stationary and multi-scale characteristics, with damage-sensitive features of different damage patterns spanning multiple temporal scales. Existing methods fail to sufficiently extract and fuse multi-scale damage-sensitive features. To this end, this paper proposes a novel Residual Dual Attention Multiscale Network (RDAMNet). The network innovatively designs a signal-level multi-scale decoupling strategy that extracts damage-sensitive features at different scales from complementary signal representations through a multi-branch differentiated architecture. Furthermore, an ECA-SE dual attention mechanism is designed to collaboratively enhance damage-related channel responses at both the feature extraction and fusion stages. Multiple independent experimental results on a publicly available dataset demonstrate that RDAMNet achieves a mean damage recognition accuracy and a weighted F1-score of 95.39% and 95.37%, respectively, significantly outperforming five compared methods. Cross-condition generalization experiments further demonstrate that RDAMNet maintains mean accuracies exceeding 94% across different wind speed and wind direction combinations, validating its stability across operating conditions. Moreover, RDAMNet only contains 663,783 parameters with a single-sample GPU inference time of 5.35 ms, exhibiting a favorable performance–efficiency trade-off. The ablation study verifies the effective contribution of each core component, and branch importance analysis, together with Grad-CAM visualization, further substantiates the multi-scale feature learning capability of the network. The proposed method provides an effective technical approach for intelligent structural health monitoring of FWTs in complex oceanic environments.