Energy harvesting with cognitive radio and nonorthogonal multiple access (NOMA) techniques offer a promising solution to enhance the spectral and energy efficiency in Internet of Things (IoT) networks. This article investigates the performance of an overlay cognitive NOMA (OCNOMA) system tailored for IoT applications. Herein, the primary network includes a primary transmitter-receiver pair, whereas the secondary network comprises an energy-constrained full-duplex (FD) relaying-based secondary transmitter (ST) with its intended multiple receivers. Accordingly, ST employs a time-switching (TS)/power-splitting (PS)-based receiver architecture to harvest the energy from the radio-frequency signal of primary transmission and, thereby, uses this energy to relay the primary signal and to transmit its own signals simultaneously using the OCNOMA principle. For this, we propose a cooperative spectrum-sharing transmission (CSST) scheme using the decode-and-forward relaying strategy, while considering the realistic assumptions of FD-based loop self-interference, NOMA-based imperfect successive interference cancelation, and the transceiver hardware impairments in IoT devices. Adopting Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> fading environments, we comprehensively analyze the performance by deriving the expressions of the outage probability for the primary and secondary networks for the FD-based CSST scheme under both TS and PS protocols. Thereby, we disclose some apposite ceiling effects and provide an insight on deciding the value of the OCNOMA power allocation factor for efficacious spectrum-sharing cooperation. We further quantify throughput and energy efficiency for the overall system. Our results demonstrate the performance advantages of the proposed FD CSST scheme over the benchmark schemes and provide useful guidelines for the practical design of cognitive IoT networks.