Looking at your video it seems you have a very nice basic cast to work with. You have what looks to be real good straight line path. That right there is a pain for many to achieve. The power application you use seems quite good if not a bit too early with max power at times.
Your haul... There is not really have much of a haul that I can see. Every so often I see your line hand going up to a great position to begin a haul but alas, you were simply gathering up line for another cast.
Try pantomiming the haul without a rod in your hand. Double Haul Basics
Remember... clean, slick line and quite a bit of line mass out past the line tip will really aid in the 'up' portion of the double haul where you have to feed line back into the cast. If line doesn't feed back easily in the 'up' portion of the double haul you're just inviting frustration.
One thing you do is worse than scratching a chalk board to me, that is stripping line while false casting as you did when you went to the 5 weight video section at about 2:40 in. The angle your fly line takes from the spool to your line hand is an extreme turn and wreaks havok on the fly line because of the extreme, tensioned turn the line has to make but also, the friction created can actually gouge divots in the line gaurd on the reel. I can show you a Hardy with a divoted line guard from a friend borrowing the reel for a short time. Strip off the line you need then cast, it could save you gear damge.