The family of Jordan-Moore-Gibson-Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations.