Kim and Winfree have synthesized a well-known network of transcriptional oscillators in vitro using a modular architecture of synthetic gene analogues and a few enzymes that, in turn, could be used to drive a variety of downstream circuits and nanodevices. However, these oscillators are sensitive to initial conditions and downstream load processes. Furthermore, the oscillations are not sustained since the inherently closed design suffers from enzyme deactivation, NTP fuel exhaustion, and waste product build up. Recently, we had shown that a partially open architecture in which a continuous-Time L1 adaptive controller, implemented inside an in silico computer that resides outside the wet-lab apparatus, can ensure sustained tunable oscillations in two specific designs of the Kim-Winfree oscillator networks. Here, we present its discrete-Time version. As before, we consider two broad cases of operation: (1) the oscillator network operating in isolation, and (2) the oscillator network driving a DNA tweezer subject to a variable load. In both scenarios, our simulation results show a significant improvement in the tunability and robustness of these oscillator networks. Our approach can be easily adopted to improve the loading capacity of a wide range of synthetic biological devices.