The bismuth double perovskite Cs₂AgBiBr₆ has been regarded as a potential candidate for lead-free perovskite photovoltaics. A detailed study on the coherent acoustic phonon dynamics in the pure, Sb- and Tl-alloyed Cs₂AgBiBr₆ single crystals is performed to understand the effects of alloying on the phonon dynamics and band edge characteristics. The coherent acoustic phonon frequencies are found to be independent of the alloying, while the damping rates are highly dependent on the alloying. Based on the mechanism of coherent acoustic phonon damping, a technique has been successfully developed that can accurately extract the absorption spectra near the indirect band gap for these single crystals with coefficients on the order of 10² cm⁻¹.