The statistical study of precision medicine is concerned with dynamic treatment regimes (DTRs) in which treatment decisions are tailored to evolving patient-level information. An optimal DTR is the sequence of treatment decisions that yields the best expected outcome. Statistical methods for optimal DTRs of observational data are theoretically complex and hardly accessible to researchers, especially when the outcome is survival time subject to right censoring. We propose a doubly-robust method, called dynamic weighted survival modeling, for estimating optimal DTRs for such endpoints. Our method is available in the DTRreg R package, thus facilitating its application by researchers. We illustrate our novel method with an application to the treatment of type 2 diabetes, estimating a DTR for second- and third-line diabetes therapies that maximizes the time until the occurrence of a major cardiovascular event in patients for whom metformin, the preferred first-line treatment, has failed.