Understanding the dynamics of compound dry-hot extremes (CDHEs) is critical for climate risk assessment in semi-arid regions. This study over the state of Rajasthan in India investigates the evolution of rainfall deficits (defined by the 30th percentile precipitation threshold, P30) and temperature extremes (defined by the 70th percentile temperature threshold, T70), and their co-occurrence over six decades (1960-2024) using regridded district-level TerraClimate reanalysis data. Extreme thresholds were identified through percentile-based indices, and both frequency and persistence of events were analyzed. A first-order Markov chain framework developed based on exiting methodology was further applied to estimate stationary and transition probabilities, providing insights into long-term likelihoods and state-to-state shifts of compound extremes. Results show a fundamental spatio-temporal shift in extremes. Long-duration rainfall deficits that previously dominated the eastern and south-eastern regions have declined and are replaced by shorter but more frequent P30 events. Simultaneously, T70 events have increased sharply in frequency and persistence across the state, leading to more common cooccurrence of CDHE conditions. The early 2000s marked a peak in CDHE frequency and mean duration, particularly across central, south-western, and south-eastern Rajasthan. Stationary probabilities confirm that rainfall-only droughts since the 2000s occur infrequently, while heat-only and compound CDHE states persisted for all study period (1960-2024). Transition probabilities further indicate that both rainfall deficits and heat extremes have a high likelihood of evolving into compound states, with strong persistence once compounding begins. These findings highlight that Rajasthan's climate is moving into a regime defined by frequent and intensified compound extremes. The shift poses adverse implications on agriculture, water resources, and livelihoods, underscoring the urgent need for designing integrated monitoring and adaptation strategies for similar semi-arid regions across the globe to address concurrent rainfall variability and rising temperatures.