<p>Weighted Poisson models provide a versatile framework for modeling underdispersed, equidispersed, and overdispersed count data. Recent advances in computation have enabled the development of increasingly elaborate families whose probability weights involve special functions, particularly gamma functions and their ratios. Motivated by this literature, we introduce a new class of flexible count models obtained by weighting the Poisson law with affine functions and, more generally, affine powers of affine functions. The proposed generalized affine distribution (GAD) family replaces special-function-based weight structures with simpler affine expressions while retaining the ability to accommodate underdispersion, overdispersion, and heavy-tailed behavior. In particular, the model captures tail behavior consistent with that of gamma-function-based constructions through its asymptotic probability ratios. Notably, a subclass admits an exponential-family representation, which enables straightforward and numerically stable inference via moment-matching methods. We establish key distributional properties of the GAD family, develop practical estimation procedures, and assess performance through extensive simulation studies. The practical usefulness of the proposed models is further demonstrated using two real-world count datasets, where the GAD family delivers competitive or improved fits relative to classical alternatives.</p>