Many epidemiological and clinical studies aim at analyzing a time-to-event endpoint. A common complication is right censoring. In some cases, it arises because subjects are still surviving after the study terminates or move out of the study area, in which case right censoring is typically treated as independent or non-informative. Such an assumption can be further relaxed to conditional independent censoring by leveraging possibly time-varying covariate information, if available, assuming censoring and failure time are independent among covariate strata. In yet other instances, events may be censored by other competing events like death and are associated with censoring possibly through prognoses. Realistically, measured covariates can rarely capture all such associations with certainty. For such dependent censoring, often covariate measurements are at best proxies of underlying prognoses. In this paper, we establish a nonparametric identification framework by formally admitting that conditional independent censoring may fail in practice and accounting for covariate measurements as imperfect proxies of underlying association. The framework suggests adaptive estimators which we give generic assumptions under which they are consistent, asymptotically normal, and doubly robust. We illustrate our framework with concrete settings, where we examine the finite-sample performance of our proposed estimators via a Monte-Carlo simulation and apply them to the SEER-Medicare dataset.