The requirement for massive connectivity in 5G and beyond 5G systems pose new challenges to random access protocol design. As a promising technique to afford massive short packet connections, unsourced random access (URA) gains traction. However, the existing URA schemes were developed for static channels and not implementable in fast fading channels. In this paper, we propose a novel URA strategy tailored to time-varying Rayleigh fading channels when the system loads and instantaneous channels are unknown to the receiver. With a time slotted transmission framework, we devise a low-complexity error correcting code and a decoding algorithm leveraging successive interference cancellation (SIC). The receiver corrects errors across slots by using SIC to decode the transmitted messages from highly interfering superposed signals. The asymptotic error rate is derived and a trade-off between the minimum required energy required for reliable communication and the user density is analyzed. Our simulation results verify that the proposed scheme achieves a near-optimal energy efficiency performance when the system load is in the moderate system load regime and a significant performance gain over the existing URA schemes in time varying fading channels.