The integration of the Industrial Internet of Things (IIoT) with Artificial Intelligence-Generated Content (AIGC) offers new opportunities for smart manufacturing, but it also introduces challenges related to computation-intensive tasks and low-latency demands. Traditional generative models based on cloud computing are difficult to meet the real-time requirements of AIGC tasks in IIoT environments, and edge computing can effectively reduce latency through task offloading. However, the dynamic nature of AIGC tasks, model switching delays, and resource constraints impose higher demands on edge computing environments. To address these challenges, this paper proposes an AIGC task offloading framework tailored for IIoT edge computing environments, considering the latency and energy consumption caused by AIGC model switching for the first time. IIoT devices acted as multi-agent collaboratively offload their dynamic AIGC tasks to the most appropriate edge servers deployed with different generative models. A model aware AIGC task offloading algorithm based on Multi-Agent Deep Deterministic Policy Gradient (MADDPG-MATO) is devised to minimize the latency and energy. Experimental results show that MADDPG-MATO outperforms baseline algorithms, achieving an average reduction of 6.98% in latency, 7.12% in energy consumption, and a 3.72% increase in task completion rate across four sets of experiments with model numbers ranging from 3 to 6, it is demonstrated that the proposed algorithm is robust and efficient in dynamic, high-load IIoT environments.