We demonstrate that shunting of superconductor-insulator-superconductor (S-I-S) Josephson junctions by superconductor-insulator-normal metal (S-I-N) structures having pronounced nonlinear IV characteristics can remarkably modify the Josephson dynamics. In the regime of Josephson generation the phase behaves as an overdamped coordinate, while in the superconducting state the damping and current noise are strikingly small, that is vitally important for application of such junctions for readout and control of Josephson qubits. Superconducting Nb/AlOx/Nb junction shunted by Nb/AlOx/AuPd junction of S-I-N type was fabricated and, in agreement with our model, exhibited nonhysteretic IV characteristics at temperatures down to at least 1.4 K.

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