Drought, as a natural disaster with wide-ranging impacts and long duration, has an adverse effect on the global economy and ecosystems. In this paper, four remote sensing drought indices, namely the Crop Water Stress Index (CWSI), Vegetation Supply Water Index (VSWI), Temperature Vegetation Dryness Index (TVDI), and Normalized Difference Water Index (NDWI), are selected for drought analysis. The correlation analysis is carried out with the self-calibrated Palmer Drought Severity Index (sc-PDSI), and based on the optimal index (CWSI), the spatiotemporal characteristics of drought in Shaanxi Province from 2001 to 2021 were studied by SEN trend analysis, Mann–Kendall test, and a center of gravity migration model. The results show that (1) the CWSI performs best in drought monitoring in Shaanxi Province and is suitable for drought studies in this region. (2) Drought in Shaanxi Province shows a decreasing trend from 2001 to 2021; the main manifestation of this phenomenon is the decrease in the occurrence of severe drought, with severe drought covering less than 10% of the area in 2010 and subsequent years. The most severely affected regions in the province are the northern Loess Plateau region and Guanzhong Plain region. In terms of the overall trend, only 0.21% of the area shows an increase in drought, primarily concentrated in the Guanzhong Plain region and the outskirts of the Qinling–Bashan mountainous region. (3) Drought conditions are generally improving, with the droughts’ center of gravity moving northeastward at a rate of 3.31 km per year. The results of this paper can provide a theoretical basis and a practical reference for drought control and decision-making in Shaanxi Province.