Regional land-use change is the leading cause of ecosystem carbon stock change; it is essential to investigate the response of LUCC to carbon stock to achieve the strategic goal of “double carbon” in a region. This paper proposes a residual network algorithm, the Residual Multi-module Fusion Network (R-MFNet), to address the problems of blurred feature boundary information, low classification accuracy, and high noise, which are often encountered in traditional classification methods. The network algorithm uses an R-ASPP module to expand the receptive field of the feature map to extract sufficient and multi-scale target features; it uses the attention mechanism to assign weights to the multi-scale information of each channel and space. It can fully preserve the remote sensing image features extracted by the convolutional layer through the residual connection. Using this classification network method, the classification of three Landsat-TM/OLI images of Zhengzhou City (the capital of Henan Province) from 2001 to 2020 was realized (the years that the three images were taken are 2001, 2009, and 2020). Compared with SVM, 2D-CNN, and deep residual networks (ResNet), the overall accuracy of the test dataset is increased by 10.07%, 3.96%, and 1.33%, respectively. The classification achieved using this method is closer to the real land surface, and its accuracy is higher than that of the finished product data obtained using the traditional classification method, providing high-precision land-use classification data for the subsequent carbon storage estimation research. Based on the land-use classification data and the carbon density data corrected by meteorological data (temperature and precipitation data), the InVEST model is used to analyze the land-use change and its impact on carbon storage in the region. The results showed that, from 2001 to 2020, the carbon stock in the study area showed a downward trend, with a total decrease of 1.48 × 107 t. Over the course of this 19-year period, the farmland area in Zhengzhou decreased by 1101.72 km2, and the built land area increased sharply by 936.16 km2. The area of land transfer accounted for 29.26% of the total area of Zhengzhou City from 2001 to 2009, and 31.20% from 2009 to 2020. The conversion of farmland to built land is the primary type of land transfer and the most important reason for decreasing carbon stock. The research results can provide support, in the form of scientific data, for land-use management decisions and carbon storage function protections in Zhengzhou and other cities around the world undergoing rapid urbanization.