In recent years, the loss of fresh food during the distribution and sales process has become a serious concern. The introduction of traceability system is effective for managing freshness. From the perspective of selling price and freshness of fresh food, it is important for related vendors to make an appropriate choice between presale or spot sale or adopt a dual sale through offline channel and online channel. A cap-and-trade policy that carbon quotas allocated to vendors by the government can be traded among vendors, is effective in reducing carbon emissions. In this paper, we consider both freshness management—including traceability—and carbon emission reduction. Specifically, this paper examines two models: Model SR, in which fresh food is sold to consumers by both a supplier and a retailer through a dual-channel food supply chain (DCFSC) that includes presale or spot sale; and Model R, in which fresh food is sold by the retailer to consumers. We propose the optimal operation of DCFSC under the supplier Stackelberg game. In the numerical analysis, this paper first compares the optimal operation of each Model of DCFSC and evaluates the social welfare using a numerical example. Then, as a sensitivity analysis, this paper clarifies the effects of (i) the ratio of presale, λ, and (ii) the ratio of offline channel, φ, on the optimal strategy, the profit, the consumer surplus and the environmental impact and the social welfare in DCFS for Model R and Model SR.