In this study, we design a reservoir computing (RC) network by exploiting short- and long-term memory dynamics in Au/Ti/MoS$_2$/Au memristive devices. The temporal dynamics is engineered by controlling the thickness of the Chemical Vapor Deposited (CVD) MoS$_2$ films. Devices with a monolayer (1L)-MoS$_2$ film exhibit volatile (short-term memory) switching dynamics. We also report non-volatile resistance switching with excellent uniformity and analog behavior in conductance tuning for the multilayer (ML) MoS$_2$ memristive devices. We correlate this performance with trap-assisted space-charge limited conduction (SCLC) mechanism, leading to a bulk-limited resistance switching behavior. Four-bit reservoir states are generated using volatile memristors. The readout layer is implemented with an array of nonvolatile synapses. This small RC network achieves 89.56\% precision in a spoken-digit recognition task and is also used to analyze a nonlinear time series equation.