Domain-specific systems-on-chip (DSSoCs) aim to narrow the gap between general-purpose processors and application-specific designs. CPU clusters enable programmability, while hardware accelerators tailored to the target domain minimize task execution times and power consumption. Traditional operating system (OS) schedulers can diminish the potential of DSSoCs as their execution times can be orders of magnitude larger than the task execution time. To address this problem, we propose a dynamic adaptive scheduling (DAS) framework that combines the advantages of a fast, low-overhead scheduler and a sophisticated, high-performance scheduler with a larger overhead. We present a novel runtime classifier that chooses the better scheduler type as a function of the system workload, leading to improved system performance and energy-delay product (EDP). Experiments with five real-world streaming applications indicate that DAS consistently outperforms fast, low-overhead, and slow, sophisticated schedulers. DAS achieves a 1.29x speedup and 45% lower EDP than the sophisticated scheduler under low data rates and a 1.28x speedup and 37% lower EDP than the fast scheduler when the workload complexity increases. Furthermore, we demonstrate that the superior performance of the DAS framework also applies to hardware platforms, with up to a 48% and 52% reduction in execution time and EDP, respectively.