This work proposes a coordinated non-orthogonal multiple access (NOMA) scheme to achieve robust covert communication in downlink cooperative NOMA systems under eavesdropping surveillance, where covert messages can be hidden flexibly by the simultaneous transmission of a public message without extra jamming sources. To assess and enhance the covertness, we first derive exact closed-form expressions for the detection error probability (DEP) of each phase transmission at the eavesdropper and then provide a guide in achieving the exact closed-form expression of the respective judgment threshold and the minimal DEP. Based on this, we derive a closed-form solution for achieving the optimal power resource to maximize the covert rate while ensuring the minimum public rate and meeting the covertness requirement. Simulation results validate our mathematical frameworks and demonstrate a significant enhancement of 0.3 bps/Hz in covert rate performance when the system targets the communication covertness of 99%.