To address access collisions caused by massive user access in networks, a dynamic uplink random access method based on non-orthogonal multiple access (NOMA) is proposed. The method incorporates an adaptive resource allocation mechanism that dynamically adjusts time slot configurations and updates the access class barring (ACB) factor according to network conditions. After each round of random access, the system records the number of idle, successful, and collided time slots. These statistics are then used to estimate the number of remaining users. The process is iteratively repeated until all users successfully access the network. To evaluate the proposed method, a complete theoretical analysis framework is established under the Nakagami-m fading channel environment, from which closed-form expressions for access success probability and system throughput are derived. Simulation results show that, compared with the traditional dynamic ACB-based NOMA random access algorithm (DNRA) and the fixed ACB-based orthogonal random-access algorithm (FORA), the proposed method improves system throughput by 30.41% and 48.22%, respectively. Furthermore, under varying user loads, the method consistently achieves higher access success probabilities and maintains high system throughput across different network conditions.