Stabilized dynamic treatment regimes are sequential decision rules for individual patients that not only adapt over the course of the disease progression but also remain consistent over time in format. The estimation of stabilized dynamic treatment regimes becomes more complicated when the clinical outcome of interest is survival time subject to censoring. We propose two novel methods, censored shared-Q-learning and censored shared-O-learning, for this purpose. Both methods incorporate clinical preferences into a qualitatively fixed rule, where the parameters indexing the decision rules that are shared across stages can be estimated simultaneously, while adjusting for censoring. We conducted extensive simulation studies, which demonstrated superior performance of the proposed method. We analyzed data from the Framingham Study using the proposed methods.