Is there some way to make them cut out the barrel-hogging (preferably besides prohibiting hunting and farming) and make booze-barreling a priority over it?
Cooking! Cooking uses barrels, but cooking can also free up barrels. Switch your kitchen from making easy meals to making lavish meals, and ensure that otherwise inedible large-stack foods like Dwarven Syrup and Quarry Bush Leaves are close to the kitchen for use as filler. Total stack size for a stack of prepared meals is the sum of the stack sizes of the ingredients used, so using large stacks of ingredients will help generate large stacks of food.
"Large stacks? That's even worse," I hear you saying. Not so! When a stack of prepared meals is larger than 25, it no longer fits in a barrel and will be stored on the ground. Because it's prepared food and stored indoors, however, it will not spoil. This gives you high-value meals which take up a single tile of your fortress without occupying a barrel.
Example:A Proficient cook can generate
1 Barrel of Cow Meat + 1 Barrel of Plump Helmet = 1 Barrel of Plump Helmet Biscuit worth ~$458 (and frees 1 Barrel)
or1 Barrel of Cow Meat + 1 Barrel of Plump Helmet + 1 Bag of Quarry Bush Leaves + 1 Barrel of Dwarven Syrup = 1 stack of Dwarven Syrup Roast worth ~$6600 (and frees 3 Barrels and a bag)
Along the same lines, consider fertilizing the fields that create your booze plants, and letting a small number of dwarves practice the Grower/Herbalist skill to maximize crop yields. The more plants per stack, the greater a volume of booze you can fit in a single barrel.