One thing I see is rust. That's no crime, we all have it at the beginning of the season and we have to ski it off.
I think mike_m may have a point about pronation. I have fairly severe pronation in my left foot and leg, but the right leg has none. One of the hall marks of that is that I shape turns differently to each side, and it is not uncommon for my turns to begin to develop sequentially (that is, both skis do not turn at the same time, as in simultaneously). In both the long and short turns, I see that you are moving sequentially most of the time.
Pronation or Supination can be addressed by good bootfitters, who will cant and align you, as needed, but begin with a custom footbed and go from there.
That was quite in evidence to me on Sunday, my first day on skis this year at Cannon. I have a new pair of Salomon XMax 100's replaced under warranty because of a defective liner. I got the fitting done quite late, and have not had the time yet to be canted and aligned, which I know from past experience that I will need. My turns were sequential, and my left ski wanted to wash in the tip without me overcompensating on pressuring the tip to the inside and leading the turn more consciously with forward and diagonal movement. With proper alignment and canting, that need is not evident.
You will want to work with an expert boot fitter and not some shop rat who just pulls boots off the rack. Hopefully, you can find someone in your area who is good, local ski instructors would be good places to search for recommendations.
Lacking that, I know that Paul Richelson in Plymouth, New Hampshire is very good. I have used him for years. He's about a mile off Rt. 93 at exit 26 - on your way to Cannon or other ski areas in central and Northern NH. It will be spendy but will last for many years and is well worth doing.
Some other notes on your skiing: In short turns, your upper body should stay squared to the fall line, and your lower body skis into separation of the two and into counter. I see little separation in your movements, and you are using upper body rotary to assist that push off your tails that others have identified. I see almost no forward movement into the new turns. Rather, your movement is up and lateral. Because most of your weight is on the inside ski, the outside ski is not properly weighted and your skis diverge - the outside ski moves away from the inside ski as the turn develops. You are banking off the inside ski and you end up well behind, so the finish of the turn is more to the tails. That requires a large movement to get forward and it does not allow you to establish your new turn before you get to the fall line. Since you are not pressuring the tips most of the time, your will be more likely to get knocked around by loose snow and variable surfaces.
I had the pleasure of having Nemesis256 in a private lesson when he came up to Cannon late last winter. He is very athletic and capable, and a better skier than the videos show him to be. One of the things we worked on was trying to balance over and pressure the outside ski more, and he was a good student. But, as the videos show, old muscle memory dies hard. He's far from alone in that, and it takes a fair amount of conscious effort to take newly learned movement patterns and turn them into muscle memory.