To gain insight into the unconventional superconductivity of Fe-pnictides with no electron pockets, we measure the thermal conductivity $kappa$ and penetration depth $lambda$ in the heavily hole-doped regime of Ba$_{1-x}$K$_x$Fe$_2$As$_2$. The residual thermal conductivity $(kappa/T)_{T rightarrow 0,{rm K}}$ and $T$-dependence of $lambda$ consistently indicate the fully gapped superconductivity at $x=0.76$ and the (line) nodal superconductivity at higher hole concentrations. The magnitudes of $frac{kappa}{T}cdot T_c|_{T rightarrow 0,{rm K}}$ and $frac{dlambda}{d(T/T_c)}$ at low temperatures, both of which are determined by the properties of the low-energy excitations, exhibit a highly unusual non-monotonic x-dependence. These results indicate a dramatic change of the nodal characteristics in a narrow doping range, suggesting a doping crossover of the gap function between the s-wave states with and without sign reversal between $Gamma$-centered hole pockets.
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