Information-bearing reconfigurable intelligent surfaces (IB-RIS) provide a promising solution to self-sustainable and green communications by harvesting ambient radio frequency energy while embedding information via passive reflection. This paper investigates a self-sustainable IB-RIS (SIB-RIS)-assisted non-orthogonal multiple access (NOMA) network operating in an underlay cognitive radio (CR) system. Specifically, a multi-antenna primary transmitter (PT) serves a primary user (PU) and concurrently illuminates the secondary nodes, which enables each SIB-RIS to perform simultaneous energy harvesting and backscatter-based information embedding at each RIS. Based on this model, a weighted sum spectral efficiency (WSSE) maximization problem is formulated for the secondary network by jointly optimizing the PT transmit beamforming vector, the SIB-RIS reflection coefficients, and the power-splitting ratios. To tackle the intricately-coupled non-convex problem, an efficient block coordinate descent (BCD) optimization framework is developed, which leverages fractional programming via Lagrangian dual and quadratic transforms together with a difference-of-convex programming approach. Numerical results demonstrate that the proposed SIB-RIS-assisted NOMA CR system yields substantial WSSE gains over both orthogonal multiple access (OMA)-based and active antenna schemes. Moreover, a 2-bit discrete-phase SIB-RIS implementation achieves competitive to which WSSE performance, confirming the practicality of the low-resolution architecture.