This study emphasizes a three-level wooden furniture supply chain, which involves one supplier, one manufacturer, and one retailer. Focusing on maximizing the profit of the supply chain while adhering to low-carbon principles, the Three-level Leader-follower Game (TLG) model, Stackelberg Game Model I (SGI model), Stackelberg Game Model II (SGlI model), and Cooperative Decision-making (CD) model were established by using game theory. The carbon emission reduction cost and benefit sharing contract was introduced into the model with the maximum profit, and the ranges of sharing coefficients for a solid wood bed supply chain and the optimal decision-making process for each supply chain member were discussed. Results showed that the profit for the solid wood bed supply chain reached maximum under the CD model, followed by the SGII model, and then the SGI model, and the TLG model showed the lowest profit. A higher preference for low-carbon products can lead to lower demand for products and higher retail prices. Through introducing the cost and benefit-sharing contract into the CD model, the profit of the supply chain can be guaranteed with different sharing coefficients, and the profit of each member was improved compared to the TLG model.