With the development of industrialization and intelligence, the Industrial Internet of Things (IIoT) has gradually become the direction for traditional industries to transform into modern ones. In order to adapt to the emergence of a large number of edge access devices such as sensors, as well as the demand for high-consumption and low-latency computing tasks, Mobile Edge Computing (MEC) has been proposed as an effective paradigm by the academic community. Users can offload tasks to MEC servers, greatly reducing the computing latency and energy consumption when using services. However, traditional single access point edge computing networks cannot meet the usage requirements of a large number of users. In addition, the privacy leakage issues arising from the offloading process are also easily overlooked. In this paper, we propose a privacy-preserving offloading scheme based on stochastic game theory considering multiple access points. We first construct a multi-access point offloading framework and quality of service (QoS) model. In terms of privacy, the privacy risks caused by the offloading preferences of different edge nodes are studied, and the privacy entropy is used to evaluate the privacy protection level. We comprehensively consider the energy consumption, latency requirements, user experience, and privacy protection of the system and formulate the problem as a Markov Decision Process (MDP). Finally, a joint optimal DRL algorithm with privacy preservation (JODRL-PP) is proposed to achieve the optimal offloading scheme of the system. Simulation results verify the effectiveness of our model and algorithm.