Drought is a phenomenon that is caused by several factors and can be divided into meteorological drought, agricultural drought, hydrological drought, and socioeconomic drought. In this study, the characteristics of propagating from meteorological drought to agricultural (or hydrological) drought in the Andong Dam basin and Hapcheon Dam basin located in the Nakdong River basin in Korea were investigated. Standardized precipitation index (SPI), standardized soil moisture index (SMI), and standardized runoff index (SRI) were used to characterize meteorological, agricultural, and hydrological droughts, respectively. SPI-m (1–12) on various timescales and SMI-1 (or SRI-1) were selected as drought propagation timeseries, such that a correlation analysis was performed to evaluate the correlation and propagation time between meteorological and agricultural (or hydrological) drought. Propagation probability was quantified using a copula-based model. The correlation between meteorological and agricultural (or hydrological) droughts was not significantly affected by seasons. A relatively strong correlation was found in summer. A relatively weak correlation was shown in autumn. In addition, it was found that there was a difference in correlation between the Andong Dam basin and the Hapcheon Dam basin. On the other hand, in both watersheds, the propagation time was as long as 2 to 4 months in spring and decreased to 1 month in summer.