To solve the problem of low straw-cutting efficiency of single-disc openers of no-till planters under conditions of high soil moisture content, a single-disc furrowing and straw-cutting device was designed based on the support-cutting principle. To improve the straw-cutting ability of the disc opener when it operates under high-moisture-content soil conditions and to make sure that the straw that is not cut by the disc coulter can be cut smoothly by the disc opener, the support shovel was designed, and the operation mechanism of the support shovel device was analyzed. The soil moisture content, the support shovel’s entry angle, the support shovel’s entry gap angle, and the support shovel’s tip margin were identified as the factors influencing the device design through the theoretical analysis of the furrowing and straw-cutting device. Through the discrete element method (DEM), a single-factor simulation test was first conducted to analyze how different soil moisture contents affected the device’s ability to cut straw, and the results showed that the number of broken bonds was lowest when the soil moisture content was 20 ± 1%, and the time taken for the straw to be wholly cut off was also the longest. Then, a quadratic orthogonal simulation test was conducted to construct a regression model and optimize the parameters at the soil moisture content of 20 ± 1%, and the results revealed that the significant order of each factor’s influence on the number of broken bonds is as follows: entry gap angle, entry angle, and shovel tip margin. In addition, the device’s overall operation quality was better when the entry angle was 49°, the entry gap angle was 0°, and the shovel tip margin was 10 mm. At this time, the number of broken bonds was predicted to be 506. Finally, the simulation validation test was run, and the number of broken bonds was obtained to be 478, with a relative error of 5.6% from the predicted value. According to the optimal parameters to complete the device trial production and field test, the results show that the average cut-off rate of the device is 71.7%, the stability coefficient of the furrowing depth is 90.87%, and the performance of the furrow opening is excellent, which meets the requirements of a no-tillage seeding operation. This study can provide a reference for the design and improvement of no-tillage seeding machines under conditions of high soil moisture content.