The statistical study of precision medicine is concerned with dynamic treatment regimes (DTRs) in which treatment decisions are tailored to patient-level information. Individuals are followed through multiple stages of clinical intervention, and the goal is to perform inferences on the sequence of personalized treatment decision rules to be applied in practice. Of interest is the identification of an optimal DTR, that is, the sequence of treatment decisions that yields the best expected outcome. Statistical methods for identifying optimal DTRs from observational data are theoretically complex and not easily implementable by researchers, especially when the outcome of interest is survival time. We propose a doubly robust, easy to implement method for estimating optimal DTRs with survival endpoints subject to right-censoring which requires solving a series of weighted generalized estimating equations. We provide a proof of consistency that relies on the balancing property of the weights and derive a formula for the asymptotic variance of the resulting estimators. We illustrate our novel approach with an application to the treatment of rheumatoid arthritis using observational data from the Scottish Early Rheumatoid Arthritis Inception Cohort. Our method, called dynamic weighted survival modeling, has been implemented in the DTRreg R package. for this article are available online.