Peer-to-peer (P2P) energy trading enables an efficient regulation of distributed renewable energy among prosumers, implicitly promoting low-carbon operation. This study proposes a novel P2P energy trading scheme with coupled electricity-carbon (E/C) market that co-optimizes both power and carbon emission flows. To facilitate the low-carbon operations in the market, we introduce a prosumer-driven carbon-aware distribution locational marginal price (PDC-DLMP) to serve as a pricing signal for the distribution system operator (DSO). To efficiently determine the optimal trading solutions, we adopt a two-layer data-driven approach. The first layer employs a reinforcement learning algorithm named multi-agent twin-delayed deep deterministic policy gradient (MATD3); the second layer uses a deep neural network (DNN) driven surrogate model, which is designed to map the PDC-DLMP signals, thereby eliminating the need for direct DSO intervention during market operation. This approach protects the physical model parameters of the distribution network and ensures multi-level privacy protection. Simulation results validate the effectiveness of the proposed P2P energy trading scheme with coupled E/C market, demonstrating its ability to achieve both reduced carbon emissions and lower operational costs for microgrid prosumers.