Hilly-billing denotes the activity of sitting around a campfire while consuming large amounts of alcohol in the form of cheap beer that may or may not involve the smoking of one herb or another. Other activities associated with hill-billying include making strange animal noises, awkward dances, and storytelling.
We'll be hill-billying in the backyard this weekend, you guys should come on over.
A stinking gizzard is a drinking establishment (that may or may not serve food items) that notoriously elicits derogatory remarks from customers pertaining to a characteristic stench that most often emanates from the male and female bathrooms (if there is a distinction) with the stench often consisting of aromas most often associated with urine, feces, vomit, sex, and regurgitated food.
Note that the noun phrase "stinking gizzard" may be preceded by the definite and indefinite determiners ('the' and 'a' respectively). "The stinking gizzard" is often used when there is a particularly salient drinking establishment that is notoriously 'a stinking gizzard'.
Hey man, we're going out to the stinking gizzard tonight. The place is rancid but at least the beer is cheap.