The escalating global desertification crisis underscores the urgency of understanding vegetation dynamics. This study presents an extended vegetation-water model that integrates local grazing and cross-diffusion, addressing the limitations of traditional models. Stability analysis and Turing instability theory are used to determine the conditions for Turing instability and bifurcation. Multiscale analysis is used to derive amplitude equations at the Turing bifurcation point, revealing the intricate vegetation pattern structures associated with different control parameters. Our results show that self-diffusion promotes diverse vegetation patterns, while cross-diffusion can potentially accelerate desertification under certain circumstances. Furthermore, we investigate the effect of grazing intensity on vegetation patterns and find that excessive grazing significantly increases the risk of desertification. To improve the realism of the model, we incorporate more accurate representations of plant-root water uptake and soil-water diffusivity. These findings not only contribute to the theoretical understanding of ecological dynamics, but also inform potential management approaches for grazing and desertification mitigation, such as region-specific grazing management and water retention measures.