This study primarily focuses on the impact of carbon emissions generated during the fresh product fresh-keeping process on the traditional fresh supply chain (FSC1) and the low-carbon fresh supply chain with emission reduction behavior (FSC2). Its objective is to investigate how these supply chains dynamically adjust the production quantity, fresh-keeping level, and emission reduction level of fresh products when constrained by the dual-carbon policy. The decision competition model between the two supply chains was explored under two scenarios: the horizontal Nash (HN) game and the long-term Stackelberg (LS) game, wherein FSC1 assumes the role of the leader. The findings indicate that as the carbon tax and carbon trading price increase, the fresh supply chain tends to decrease fresh-keeping intensity while increasing investment in emission reduction. Carbon emissions generated during the fresh product fresh-keeping process have a smaller impact on FSC2's profit than on FSC1, and FSC2 exhibits stronger adaptability under the dual-carbon policy. Irrespective of the game structure, if the adjustment speed of FSC1's production decision exceeds the stable range, its own profit will decrease while the other party's profit will increase. However, if the adjustment speed of FSC2's production decision surpasses the stable range, it will lead to a decrease in profits for both sides. Interestingly, in the LS game, when the adjustment speed of FSC1's production decision exceeds the stable range, FSC1 demonstrates a leader advantage and experiences a rapid increase in profit as the adjustment speed of FSC2's production decision increases, this phenomenon suggests that in the LS game, the leader prevails amidst market chaos.