Previous commenters have offered some excellent advice. I definitely agree that rushing and getting too deep can cause them to hang their legs or jump "over their shoulder". Horses need time on takeoff to sit back on their haunches, round their back and lift their knees up and forward over a jump, and when they are rushing or are taking off deep, they have less time to prepare themselves, and that where you end up with the flat/awkward jump/hanging knees/bad form, etc.
In the first photo, it's hard to use as a gauge because she hasn't left the ground completely. A half second later her legs would have been more square and lifted, and she probably would have looked alright. The second photo looks a tad awkward, either she is compensating for the slightly deep distance or was a bit impressed by the jump. The third I can definitely see clearly what you are talking about.
Grids and bounces can help tighten up a horse's front end. Also placing guide poles in an upside down V right near the middle of the jump also helps encourage them to snap their knees up, hopefully this photo link works to show what I am talking about
Obviously working on teaching her to relax and slow down will improve things a lot, only when she actually relaxes can she begin to use herself properly. And I know how difficult and how much on an ongoing thing that can be with young TB's!! Gotta keep working hard at it though, it will come!!
Another thing is that she may just not be physically built to be able to get her knees up nice and tight. In the pics she is not really using herself like I'd like to see, whether this is just because the jumps are small and she's not really trying or maybe because the saddle was/is hindering her, I'm not sure. She is basically not using her shoulder at all over the fence, just kind of folding her actual leg out of the way. She should be swinging her elbow and forearm up and forward, and her shoulder blade should swing back over the fence. I know you said you got a new saddle, so that might make a difference over time. Getting a good chiropractor out for an adjustment would not be a bad idea, if you haven't yet. She might have some lingering issues from the ill-fitting saddle. In my experience, anything off the track usually has at least some minor, if not major alignment issues. I know my mare STILL gets locked up in her shoulders sometimes, and I can tell that she needs an adjustment when she starts jumping more like a hunter, square knees but lower legs perpendicular to the ground, instead of her usually super tight folded up knees.
I can't watch the videos on this computer, so I can't comment about any of that. Hope this helps though!