Some type III bursts are observed to undergo sudden flux modifications, e.g., reductions and intensifications, when type III beams cross shocks in the upper corona or solar wind. First simulations are presented for type III bursts perturbed by weak coronal shocks, which type III beams traverse. The simulations incorporate spatially localized jumps in plasma density and electron and ion temperatures downstream of a shock. A shock is predicted to produce significant modulations to a type III burst: (1) a broadband flux reduction or frequency gap caused by the shock's density jump, (2) a narrowband flux intensification originating from where the downstream plasma density locally has a small gradient, (3) a possible intensification from the shock front or just upstream, and (4) changes in the frequency drift rate profile and the temporal evolution of radiation flux at frequencies corresponding to the shocked plasma. The modulations are caused primarily by fundamental modifications to the radiation processes in response to the shocked density and temperatures. The predicted intensifications and reductions appear qualitatively consistent with the available small number of reported observations, although it is unclear how representative these observations are. It is demonstrated that a weak shock can cause an otherwise radio-quiet type III beam to produce observable levels of narrowband radio emission. The simulations suggest that type III bursts with frequency-time fine structures may provide a tool to probe shocks in the corona and solar wind, especially for weak shocks that do not radiate by themselves.