In this paper, a comprehensive, free, and competitive framework for peer-to-peer (P2P) energy markets has been designed. This framework aims to promote clean energy exchanges and to reduce the social and environmental impacts resulting from carbon emissions. The proposed framework introduces a decentralized multi-market model for energy and carbon trading. By employing an energy-labeling mechanism, this framework takes the evaluation of exchanges beyond simple pricing and is designed to maximize the economic benefits of market participants while encouraging them to make greater use of renewable energy resources. By integrating the social cost of carbon (SCC) into the objective functions of market participants, the model provides incentives to reduce reliance on the upstream grid (the source of carbon emissions) and increase local clean energy exchanges. Moreover, local sellers obtain additional income by participating in the carbon certificate market, in which the certificates are issued in proportion to the amount of clean energy traded. Additionally, in this paper, a total social welfare (TSW) index has been developed. This index, with the aim of addressing the deficiencies present in traditional price-based assessments, is used as a metric for measuring the sustainable value of energy. This is supported by a realistic utility function for market participants that harmonizes economic and environmental objectives. The simulation results confirm the effectiveness and feasibility of the proposed market, which is evident through a 34.79% increase in local energy exchanges, along with a reduction in reliance on the upstream grid and a 25.66% decrease in carbon emissions.