Random reshuffling techniques are prevalent in large-scale applications, such as training neural networks. While the convergence and acceleration effects of random reshuffling-type methods are fairly well understood in the smooth setting, much less studies seem available in the nonsmooth case. In this work, we design a new normal map-based proximal random reshuffling (norm-PRR) method for nonsmooth nonconvex finite-sum problems. We show that norm-PRR achieves the iteration complexity ${\cal O}(n^{-1/3}T^{-2/3})$ where $n$ denotes the number of component functions $f(\cdot,i)$ and $T$ counts the total number of iterations. This improves the currently known complexity bounds for this class of problems by a factor of $n^{-1/3}$ in terms of the number of gradient evaluations. Additionally, we prove that norm-PRR converges linearly under the (global) Polyak-Łojasiewicz condition and in the interpolation setting. We further complement these non-asymptotic results and provide an in-depth analysis of the asymptotic properties of norm-PRR. Specifically, under the (local) Kurdyka-Łojasiewicz inequality, the whole sequence of iterates generated by norm-PRR is shown to converge to a single stationary point. Moreover, we derive last-iterate convergence rates that can match those in the smooth, strongly convex setting. Finally, numerical experiments are performed on nonconvex classification tasks to illustrate the efficiency of the proposed approach.