This paper investigates the fundamental covert rate performance in a D2D-enabled cellular network consisting of a cellular user Alice, a base station BS, an active warden Willie, and a D2D pair with a transmitter Dt and a full-duplex receiver Dr. To conduct covert communication between Alice and BS, the full-duplex Dr transmits jamming signal to confuse the active Willie and also receives signal from Dt simultaneously. With spectrum sharing, Dt can operate over either an underlay mode reusing cellular spectrum or an overlay mode using dedicated spectrum. With power control, Dr can send jamming signal to confuse Willie's detection of the transmission from Alice. We first provide theoretical results for the outage probabilities of the cellular and D2D transmissions, the average minimum detection error probability at Willie, and the achievable covert rate from Alice to BS. We then explore the power control for covert rate maximization (CRM) under the underlay mode, as well as the joint designs of power control and spectrum partition for CRM under the overlay mode. We further consider a mode selection that flexibly switches between these two modes with a probability, and also investigate the covert rate modeling and joint designs of power control, spectrum partition and mode selection probability for CRM. Finally, numerical results are presented to illustrate the covert rate performances of the network under the underlay mode, overlay mode and mode selection.