Decarbonization of the whole energy chain has been recognized as a measure to tackle the global challenge of climate change, and significant progress has already been made on the generation side to integrate renewable energy. However, the demand side is the single largest underlying factor in shaping decarbonization roadmap. Hence, the carbon emission cost should also be shared by the users according to their power consumption. In this paper, a joint electricity-carbon trading framework is designed to reduce the carbon emission through trading and demand response. A delayed carbon emission liability settlement for asynchronous markets is proposed to ameliorate the users' optimal decision from single-point optimization to interval-based optimization. To develop the optimal strategy of trading within the proposed mechanism, an improved proximal policy optimization (PPO) algorithm based on Monte Carlo reward sampling is applied. Simulation studies reveal that, compared with the market without carbon trading and users without delayed settlement, the proposed mechanism has achieved a carbon emission reduction by 40.7% and 12.7% respectively. Simulations also show the algorithm's training efficiency can be significantly improved with the proposed Monte Carlo sampling method.