This paper introduces the Sine New X–Lindley (SNXL) distribution, a new one-parameter lifetime model built by applying a sine transformation to the cumulative distribution function of the New X–Lindley distribution. The model is flexible for positive, skewed data and adds a useful option to reliability and survival analysis. We derived its main properties, including the density, distribution, survival and hazard functions, quantile function, moments, and order statistics, and also studied features such as stochastic ordering and tail behavior. We examined parameter estimation using six methods: maximum likelihood, ordinary least squares, Anderson–Darling, Cramér–von Mises, least squares, and the method of moments. Their performance was compared through a Monte Carlo simulation using bias and mean-squared error. We also extended the model to a fuzzy reliability setting by treating the scale parameter as a fuzzy number and obtaining explicit $\alpha$-cut forms for reliability and mean time to failure. To show its practical value, we applied the SNXL distribution to three real datasets on daily precipitation, heavy precipitation, and household income. In all three cases, it provided the best overall fit among the competing models under the reported goodness-of-fit measures. These results show that the SNXL distribution is a simple and effective model for skewed positive data in reliability, environmental, and economic applications.