I second Tyler's suggestions, but for long term sustained improvement I would add one more thing. I was particularly interested in looking at your transition as well as your shoulders at the top of the backswing.
With a driver you're not going to have as much tilt in your shoulders as you would with say a short iron, but even so your shoulders strike me as too level. The impact of this is apparent in the transition as you go into premature extension of your body. This standing up means you are crowding the ball and eliminating a good deal of the space that you create at address and maintained through much of the backswing -- the space between you and the ball. That space is important because if you maintain it, there is lots of room not only to get your arms in front of your body but also to do so on a more inside path -- which I think Tyler rightly points out is best understood in terms of keeping the clubhead inside the hands.
A chain reaction occurs as a result of early extension. Losing space for the arms to travel on an inside path; the right elbow getting stuck behind the right hip; an inability to do the wipe; that is, to get that right elbow moving across your body. the path heading a bit outside in to get around your body. And so on. Now for many people this would lead them to stall and flip and thus fighting big hooks and smothered shots. But you take a different approach because you continue to rotate reasonably well and definitely do not slide. Now with the outside in path and a flip you would hit smother hooks and pull hooks all day. So to combat that, you actually hold the face open and you end up with a clubface open to the path leading to anything from a pull fade to a pull slice.
The one thing I would add to what Tyler said then is that long term you will have to fight some early extension along with the other pieces Tyler recommends. Then you would find yourself even rotating through the ball more aggressively with no fear of pull hooks even though you will be entering impact with a squarer club face. You will be doing so from an inside path (the shallow) balanced by an aggressively rotating body (steepening move).
I would have liked to see a swing with a mid iron and another with a short iron to see if flat shoulder plane and early extension appear only in the driver or throughout the bag.
That's my 2 cents. Interested to see what Tyler thinks.
Hopefully this is helpful.
Jules