Full-duplex (FD) wireless technology enables a radio to transmit and receive on the same frequency band at the same time, and it is considered to be one of the candidate technologies for the fifth generation (5G) and beyond wireless communication systems due to its advantages, including potential doubling of the capacity and increased spectrum utilization efficiency. However, one of the main challenges of FD technology is the mitigation of strong self-interference (SI). Recent advances in different SI cancellation techniques, such as antenna cancellation, analog cancellation, and digital cancellation methods, have led to the feasibility of using FD technology in different wireless applications. Among potential applications, one important application area is dynamic spectrum sharing (DSS) in wireless systems particularly 5G networks, where FD can provide several benefits and possibilities such as concurrent sensing and transmission (CST), concurrent transmission and reception, improved sensing efficiency and secondary throughput, and the mitigation of the hidden terminal problem. In this direction, first, starting with a detailed overview of FD-enabled DSS, we provide a comprehensive survey of recent advances in this domain. We then highlight several potential techniques for enabling FD operation in DSS wireless systems. Subsequently, we propose a novel communication framework to enable CST in DSS systems by employing a power control-based SI mitigation scheme and carry out the throughput performance analysis of this proposed framework. Finally, we discuss some open research issues and future directions with the objective of stimulating future research efforts in the emerging FD-enabled DSS wireless systems.