In 5G smart cities, edge computing is employed to provide nearby computing services for end devices, and the large-scale models (e.g., GPT and LLaMA) can be deployed at the network edge to boost the service quality. However, due to the constraints of memory size and computing capacity, it is difficult to run these large-scale models on a single edge node. To meet the resource constraints, a large-scale model can be partitioned into multiple sub-models and deployed across multiple edge nodes. Then tasks are offloaded to the edge nodes for collaborative inference. Additionally, we incorporate the early exit mechanism to further accelerate inference. However, the heterogeneous system and dynamic environment will significantly affect the inference efficiency. To address these challenges, we theoretically analyze the coupled relationship between task offloading strategy and confidence thresholds, and develop a distributed algorithm, termed DTO-EE, based on the coupled relationship and convex optimization. DTO-EE enables each edge node to jointly optimize its offloading strategy and the confidence threshold, so as to achieve a promising trade-off between response delay and inference accuracy. The experimental results show that DTO-EE can reduce the average response delay by 21%-41% and improve the inference accuracy by 1%-4%, compared to the baselines.