Fluctuating hydrodynamics has been successfully combined with several computational methods torapidly compute the correlated random velocities of Brownian particles. In the overdamped limitwhere both particle and fluid inertia are ignored, one must also account for a Brownian drift term inorder to successfully update the particle positions. In this paper, we present an efficient computationalmethod for the dynamic simulation of Brownian suspensions with fluctuating hydrodynamics thathandles both computations and provides a similar approximation as Stokesian Dynamics for diluteand semidilute suspensions. This advancement relies on combining the fluctuating force-couplingmethod (FCM) with a new midpoint time-integration scheme we refer to as the drifter-corrector(DC). The DC resolves the drift term for fluctuating hydrodynamics-based methods at a minimalcomputational cost when constraints are imposed on the fluid flow to obtain the stresslet correctionsto the particle hydrodynamic interactions. With the DC, this constraint needs only to be imposed onceper time step, reducing the simulation cost to nearly that of a completely deterministic simulation.By performing a series of simulations, we show that the DC with fluctuating FCM is an effective andversatile approach as it reproduces both the equilibrium distribution and the evolution of particulatesuspensions in periodic as well as bounded domains. In addition, we demonstrate that fluctuatingFCM coupled with the DC provides an efficient and accurate method for large-scale dynamicsimulation of colloidal dispersions and the study of processes such as colloidal gelation.