Smart grid (SG) systems enhance grid resilience and efficient operation, leveraging the bidirectional flow of energy and information between generation facilities and prosumers. For energy demand management (EDM), the SG network requires computing a large amount of data generated by massive Internet-of-things sensors and advanced metering infrastructure (AMI) with minimal latency. This paper proposes a deep reinforcement learning (DRL)-based resource allocation scheme in a 6G-enabled SG edge network to offload resource-consuming EDM computation to edge servers. Automatic resource provisioning is achieved by harnessing the computational capabilities of smart meters in the dynamic edge network. To enforce DRL-assisted policies in dense 6G networks, the state information from multiple edge servers is required. However, adversaries can "poison" such information through false state injection (FSI) attacks, exhausting SG edge computing resources. Toward addressing this issue, we investigate the impact of such FSI attacks with respect to abusive utilization of edge resources, and develop a lightweight FSI detection mechanism based on supervised classifiers. Simulation results demonstrate the efficacy of DRL in dynamic resource allocation, the impact of the FSI attacks, and the effectiveness of the detection technique.