i clean water bowls (i actually use zoomed food dish's, theyre shallow so easy access for bd) then put in tank and fill up, they like the movement, that sometimes encourages them to try it out.
if the lizard is thirsty it will drink, theyre not silly, and or sit in the water bowl. i think they suck water up their vent.
misting is a thing of the past around here. they eat well so probably get alot of the water needs from the food.
as for brumating, just make sure it has a hide down the cooler end of tank. if its staying in their alot, let it be, then wind hours back to no more than 10hours a day. then drop temps once your sure its brumating. I actually turn lights off altogether after theyve been asleep for a week or so. if unsure just leave basking light on, especially if its eaten in the past week.
you dont need to encourage brumating, they will just do it when ready, your job is to let it be.