This research work presents a unified Bayesian hierarchical framework that implements and compares global-local shrinkage priors in logistic regression and negative binomial regression. The key feature of the approach is a representation of the likelihood using a Polya-gamma data augmentation approach that can be naturally integrated with several Bayesian priors, explicitly focusing on the Horseshoe, the Dirichlet Laplace, and Double Pareto priors. Posterior inference schemes based on Gibbs sampling were developed for both low and high-dimensional settings. Extensive simulation studies were conducted to assess the performances of these priors under different settings of sample sizes, parameter values, and covariate dimensions. The results show excellent predictive performance in terms of accuracies in most of the simulation scenarios. The method was applied to real datasets emerging from a wide range of applications. Specifically, the performance of the method was successfully validated in the context of gene-treatment interaction models that are used for assessing patient sensitivity in clinical trials. A user-friendly R package with R Shiny interface was built.