We study the quasiparticle lifetime in two- and three-dimensional Fermi liquids in two different regimes, , in the Fermi-liquid phase and in the vicinity of a density-wave transition at a finite wave vector $q_c$. Using the G${}_0$W approximation with RPA-screened interactions, we reproduce the standard behavior of the quasiparticle lifetime as $1/{\xi }_k^2$ in three dimensions and $1/[{\xi }_k^{2}ln(2/|{\xi }_k|)]$ in two dimensions, with exact prefactors for the normal Fermi-liquid phase. We show that near the transition and at the quantum critical point, the quasiparticle lifetime exhibits non-Fermi-liquid and dimension-independent scaling $\propto 1/\sqrt{|{\xi }_k|}$ in both dimensions. This dimension-independent law originates from the unusually large phase volume of critical fluctuations. Because of the existence of a soft bosonic mode with a degenerate minimum at a finite momentum $q_c$, the scattering phase space forms a circle or a spherical shell of radius $q_c$ in two and three dimensions, respectively. Consequently, the interaction between the soft bosonic mode and low-energy excitations leads to the suppression of the quasiparticle lifetime. To verify the accuracy of our analytic results, we numerically calculate the quasiparticle lifetimes of Rydberg-dressed Fermi liquids and compare them with the analytic predictions. The numerical and analytic results show excellent agreement. These results provide a clear experimental signature that can be directly observed in ultracold quantum gases via radio-frequency spectroscopy.