We analyze families of primal high-order hybridizable discontinuous Galerkin (HDG) methods for solving degenerate (second-order) elliptic problems. One major trouble regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptic in the rest. This problem is thus quite delicate to solve since the exact solution is discontinuous at interfaces separating both elliptic and hyperbolic parts. The proposed HDG method is developed in a unified and compact fashion. It can efficiently handle pure diffusive or advective regimes and intermediate regimes that combine the above mechanisms for a wide range of P\'eclet numbers, including the delicate situation of local evanescent diffusion. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is then considered. A $\theta$-upwind-based scheme is favored for the hyperbolic region, and an inspired Scharfetter--Gummel-based technique is preferred for the elliptic region. The well-posedness of the HDG method is also discussed by analyzing the consistency and discrete coercivity properties. Extensive numerical experiments are finally considered to verify the model's robustness for all the abovementioned regimes.