Accurate characterization of drought and its lagged impacts on vegetation is essential for ecohydrological studies in arid regions; however, integrated indicators capturing meteorological, agricultural, and hydrological drought simultaneously remain limited. Using the northern slope of the Tianshan Mountains (2000-2024) as a case study, we developed a composite drought index (CDI) by integrating MODIS-derived vegetation and land surface temperature, precipitation, and GRACE-GLDAS groundwater storage anomalies. Trend analysis and change-point detection were applied to examine drought evolution and delayed NDVI responses. Results show that (1) vegetation cover improved overall but exhibited spatial heterogeneity, with local degradation in intense human-activity zones; (2) single-factor indices displayed distinct spatiotemporal patterns, including stronger surface drought in the north and persistent groundwater decline; (3) the CDI correlated strongly with individual indices (mean r = 0.74, p < 0.01) and identified major historical droughts with 83% accuracy; and (4) vegetation responded to precipitation, groundwater, and temperature with mean lags of 3.4, 7.2, and 9.6 months, respectively, while 53.6% of pixels showed significant NDVI declines within 1-2 seasons under compound drought. The CDI provides an improved integrative framework for drought monitoring, supporting ecological and water-resource management in arid mountain-oasis systems.