Single-signal analysis is typically used to monitor the condition of oil-immersed transformers (OITs) for post-fault diagnosis. However, existing methods lack sensitivity to detect early-stage operational deviations that precede equipment failures. This paper focuses on proactive health monitor and detects pre-fault anomalies through multi-physics signal analysis. Current, oil temperature, and box amplitude of OITs are systematically analyzed considering their cross-domain interactions. A novel Transformer-Kolmogorov-Arnold networks (Transformer-KAN) architecture is developed to predict OIT electrical-thermal-vibration signal, and detect pre-fault anomalies. First, the cross-correlation analysis method is used to accurately quantify the delay time between OIT current, oil temperature, and vibration signals. Then, a prediction model is constructed based on Transformer self-attention mechanism and KAN nonlinear feature decomposition capability. Finally, reconstruction error is introduced to jointly optimize the OIT pre-fault anomaly detection model with the prediction error. The prediction task and anomaly detection task are combined. An on-site 500 kV OIT is used as a case study. The results show that the proposed method can significantly enhance the prediction accuracy of multi-dimensional OIT operation data and effectively detect pre-fault anomalies, providing support for OIT health monitoring and early warning.