Yes, if they are thirsty they WILL drink. If you don't train your horse he will rush to the water and can knock you out of the way in very close quarters. I am their caretaker. If I get hurt, while on the road, nobody gets watered. Period.
They get good at trailering manners by trailering short trips. My regular trip to the farrier is about one hour. I DEMAND perfect manners, and I'll take as long as it takes to wrap, load, hook to a tie, unload, unwrap, etc. Sometimes the only time a horse travels in a trailer is when you move.