Recent advancements in video diffusion models have significantly enhanced audio-driven portrait animation. However, current methods still suffer from flickering, identity drift, and poor audio-visual synchronization. These issues primarily stem from entangled appearance-motion representations and unstable inference strategies. In this paper, we introduce ConsistTalk, a novel intensity-controllable and temporally consistent talking head generation framework with diffusion noise search inference. First, we propose an optical flow-guided temporal module (OFT) that decouples motion features from static appearance by leveraging facial optical flow, thereby reducing visual flicker and improving temporal consistency. Second, we present an Audio-to-Intensity (A2I) model obtained through multimodal teacher-student knowledge distillation. By transforming audio and facial velocity features into a frame-wise intensity sequence, the A2I model enables joint modeling of audio and visual motion, resulting in more natural dynamics. This further enables fine-grained, frame-wise control of motion dynamics while maintaining tight audio-visual synchronization. Third, we introduce a diffusion noise initialization strategy (IC-Init). By enforcing explicit constraints on background coherence and motion continuity during inference-time noise search, we achieve better identity preservation and refine motion dynamics compared to the current autoregressive strategy. Extensive experiments demonstrate that ConsistTalk significantly outperforms prior methods in reducing flicker, preserving identity, and delivering temporally stable, high-fidelity talking head videos.