I'm not an English rider, but I'd say there are two problems here, both of which you've identified:
1: Chair seat.
2: Exaggerated posting.
#1 tends to cause #2, because your hips need to come so far forward before you feel balanced.
Sorry about the quality of my screen capture, but here is a shot near the beginning of your video:
I made my best guess about where your stirrup bars are and drew a vertical line. Sorry about the arrow over your face...didn't notice it at the time, but your face isn't riding the horse so...
As a rule of thumb, for a forward seat, your balance should be over the stirrup bars. About half of you in front, and half behind. Also, in a saddle that fits you, your rump should be in the lowest part of the saddle and your stirrup straps should be straight down - that is the position gravity wants you to be in.
I've never ridden in a dressage saddle, but I've read the stirrup bars are closer to the rear on one.
When half of you is in front and half behind, you are balanced over your stirrups. Then posting is just your horse pushing you up & your body UNFOLDS. Then your thighs resist gravity a little and your body folds up again, bring you down lightly.
It is possible to post in a chair seat, but you have to get used to the 'up' position being well behind the balance point of your stirrups, or you need to lean WAY forward.
My Aussie-style saddle puts me in a chair seat, and that is fine for the sort of riding I do, but it means I either need to post while leaning forward noticeably (top pic) or do so very discretely and accept that my weight will still be a bit behind the horse's center of gravity (bottom picture):
You might check your saddle fit. If it is OK, then you might be shoving your feet forward as a defense against unexpected stops or slowing, and then posting off of that. It is hard to see in the video because of the size of the video. Ask your instructor about lengthening your stirrups. It might help. Then write down the correct spot.
Someone who rides English may be able to give more accurate advice. You are doing a heck of a lot better than I was at the same level of experience.