Continuous and discrete probability models are fundamental tools for analyzing lifetime data in reliability engineering, survival analysis, and various other applied statistical fields. In this study, we propose two flexible versions of the moment exponential distribution to analyze skewed and heavy-tailed datasets. The proposed models include a continuous distribution, referred to as the Type-I Heavy Tailed moment exponential (TI-HTME) distribution, and a discrete distribution, referred to as the discrete TI-HTME distribution. Key structural mathematical properties of the TI-HTME distribution are derived, including moments, mean deviation, Bonferroni and Lorenz curves, moment-generating and characteristic functions, quantile function, Rényi, Havarda-Charvat, Tsallis, Arimoto entropies, and several reliability measures. Model parameters are estimated using a maximum likelihood estimation approach. A comprehensive Monte Carlo simulation study is conducted to assess the finite-sample performance of the proposed estimators under different sample sizes. The applicability of the proposed TI-HTME and DTI-HTME distributions is illustrated through applications to real-life data sets. The results indicate that the proposed models provide a superior fit to four datasets when compared with several competing distributions.