Optical trapping of alkali atoms such as rubidium (Rb) and cesium (Cs) underpins many modern quantum technologies, including atomic clocks, quantum information processing, and precision spectroscopy. While J-dependent polarizabilities and magic wavelengths of the $5S_{1/2}\rightarrow5P_{1/2,3/2}$ transition in Rb and the $6S_{1/2}\rightarrow6P_{1/2,3/2}$ transition in Cs are well studied, significant gaps remain in understanding hyperfine (F-dependent) polarizabilities and vector contributions. These effects are crucial for hyperfine qubits, where differential light shifts introduce decoherence, and for optical clocks, where vector polarizability can limit accuracy at the $10^{-16}$ level. We employ a relativistic all-order (AO) single–double (SD) method to calculate highly accurate dipole matrix elements and hyperfine-dependent wavefunctions. Both static ($\omega = 0$) and dynamic polarizabilities near the D1 and D2 lines are analyzed to identify magic wavelengths. For Rb $5S_{1/2}(F=1,2)$ and Cs $6S_{1/2}(F=3,4)$ ground states, we present complete sets of scalar, vector, and tensor polarizabilities. The dominant contributions arise from $5P$ and $6P$ states, while core and tail terms are small but non-negligible. Our calculations reveal strong cancellations in tensor components, large resonance-driven variations in dynamic polarizabilities, and precise magic wavelengths for both species. Vector components, particularly in Cs, were found to significantly alter trapping conditions. This work establishes a comprehensive framework for F-dependent polarizability calculations in alkali atoms. By combining relativistic all-order methods with full vector light-shift treatment, we provide benchmark data enabling precision optical trapping, state-insensitive magic wavelengths, and improved control for quantum simulation, metrology, and quantum information applications.