The dynamics of thermally-isolated Z-pinches carrying power law in time total currents (${\sim} t^S$) in magnetized resistive plasmas is studied. Time–space separable self-similar solutions with cylindrical symmetry are considered. The non-dimensional variables are chosen in a way that makes the problem consistent with the moderately resistive magnetohydrodynamic (MHD) model. For $S\,{=}\,{-}\frac15$ and Lundquist number $Lu\,{>}\, 1.5$ a non-equilibrium solution is obtained in addition to the conventional solutions for either exact, $S\,{=}\,{\pm}\frac13$, or asymptotic, $Lu\,{=}\,\infty$, equilibria (the latter is homogeneously valid for long times only if $S\,{>}\,{-}\frac15$). The problem is treated asymptotically for high dimensionless thermal-conductivity, which is proportional to the square root of the ion/electron mass ratio. To obtain a closure condition for the leading-order isothermal solution, the first-order terms in the energy equation are invoked. Radial profiles are found explicitly which depend on $S$ for equilibrium, and on $Lu$ for non-equilibrium solutions. The multiplicity of the self-similar solutions is investigated.