The rapid development of e-commerce has enabled an increasing number of people to choose online shopping. Consequently, many companies have adapted to this trend by developing online channels. This paper examines two distinct dual-channel models. The first is the manufacturer dual-channel model, in which Haier has developed its own logistics system to manage online channel distribution alongside supplying products to retailers for offline sales. The second is the retailer dual-channel model, where Jingdong has established numerous warehouses nationwide and has opened physical stores for offline product sales. Factors such as delivery time and the environmental friendliness of products significantly influence consumer purchasing decisions. The government has introduced a carbon cap-and-trade policy(cap-and-trade policy) to reduce corporate carbon emissions. This policy, along with consumer channel preferences, consumer sensitivity to delivery time, and corporate carbon emission reduction effort level, complicates dual-channel supply chain operations. Understanding how dual-channel supply chain parties make optimal decisions under the influence of the above factors and how manufacturers′ and retailers′ decisions differ in the two dual-channel models is important.This paper investigates carbon emission reduction and optimal decision-making problems in two dual-channel models and discusses the influence of relevant parameters on the decision-making of supply chain members. It considers a two-tier supply chain comprising a manufacturer and a retailer under a cap-and-trade policy. The Stackelberg game is used to model the decision-making process of supply chain members who are sensitive to both delivery time and carbon emission reduction. Consumers can find out the specific carbon footprint of the product through the carbon label on the product and the delivery timeframe through the promised delivery time of the product. The study compares carbon emission reduction efforts, delivery times, and the profits of retailers and manufacturers under scenarios with and without a carbon tax policy. It also analyzes the effect of carbon trading prices on retailers’ optimal delivery time in the retailer dual-channel model and the effect of consumers’ sensitivity to delivery time on the manufacturer’s optimal carbon emission reduction effort level. Numerical analysis is used to investigate the optimal total profit size with and without carbon tax policy for the two dual-channel models, comparing the optimal profit of retailers and manufacturers under both dual-channel models.The findings of this study suggest that 1) the cap-and-trade policy can encourage manufacturers to reduce product carbon emissions. However, the optimal delivery time length depends on the balance between the potential market demand and the cost of investing in carbon emission reduction. 2) The cap-and-trade policy is more beneficial to retailers when the investment cost for reducing carbon emissions is low and more favorable to manufacturers when the carbon emission allowance is high. 3) The “free-ride” phenomenon in the retailer dual-channel model negatively impacts the profits of both parties; therefore, contracts should be established to mitigate these cross-effects. 4) In the manufacturer dual-channel model, when optimal decisions are made, manufacturers exhibit greater efforts in reducing carbon emissions and higher aggregate demand compared to the retailer dual-channel model within the range of consumer preferences for offline channel.This study provides several implications for manufacturers, retailers, and the government. Manufacturers should consider consumer sensitivity to delivery time and carbon emission reduction to maximize profit. Retailers and manufacturers must address the “free-ride” phenomenon in the retailer dual-channel model by signing contracts to reduce the impact of cross-effects. The cap-and-trade policy yields better results when applied in the manufacturer dual-channel model. It encourages manufacturers to increase their carbon emission reduction efforts in both models, but the size of carbon credits should be set judiciously to safeguard manufacturers’ profits. Overall, this study reveals the complex and critical issues of carbon emission reduction and optimization decisions in dual-channel supply chains, offering valuable insights for practitioners and policymakers.