The stability of short-wavelength rotating modes is analysed with emphasis on their electromagnetic interaction with a nearby resistive wall. These are the edge modes such as and similar to the edge harmonic oscillations observed in the DIII-D (Burrell et al 2013 Nucl. Fusion53 073038) and other tokamaks. The existing approaches consider them assuming the wall to be ideal or disregarding the wall effect. Here, on the contrary, the wall is treated as a real conductor that serves as an energy sink. Then, in the regime where the resistive dissipation increases with the mode rotation, a dynamically maintained state with a given level of energy loss can be achieved at a given perturbation amplitude. In this model, the edge harmonic oscillation-like oscillations are described as the rotationally stabilized fast resistive wall modes (RWMs), though with larger m (poloidal wave number) compared to conventional RWMs. The cylindrical dispersion relation accounting for skin effect in the wall is solved numerically for such modes with m from 2 to 10, and the results are compared with the predictions of earlier analytical thin-wall and thick-wall theories. The computations cover a gap between them, prove their applicability in wider ranges than implied by the derivation orderings and provide better quantitative background for stability analysis. It shows that the combined effect of the mode rotation and the resistive dissipation in the wall can be an essential element in the dynamics of the edge oscillations with frequencies from several to tens kHz in tokamaks.