I've noticed that most big monsters I've fought so far have started drooling late in the fight. Is this just a sign that it's close to dead, or does it mean something specific?
All monsters have a hidden stamina bar, which they need to use for performing attacks (similar to the one you have). Much like the stamina bar the player has the maximum value of the stamina bar will decrease over time, and when the bar gets really low the monster will start drooling. During this time you'll notice them attacking less, and often messing up their attacks. They will often stand still exhausted for several seconds at a time.
Eventually the monster will go somewhere to regain their stamina, usually by eating something (although not always). For example, when Barioth is low on stamina he'll attack and eat herbivores. During this time they are completely open to attack, and by dealing enough damage you can actually stop them from finishing eating, resulting in them not regaining their stamina.
It's actually possible to drain a monsters stamina faster with Exhaust damage. This type of damage will drain a monsters stamina when dealt to any part of the monster except the head (if hit in the head it will deal KO damage instead).
Ways to deal exhaust damage: