Holy cow your horse is the spitting image of my Puck! Moves exactly the same way she's moving in that first video. She's reaching nicely into the contact, but doesn't really want to carry herself anymore than she has to. I agree with everything equinealways suggested. I throw in lots of transitions, both within gaits and between gaits, and half halts. I recently did a clinic where the clinician pointed out something that should be obvious, but it's one of those things we all have to be reminded of. Say you want to canter 100 strides. You can do the whole 100 strides in one canter, just go along for the ride, and it's never going to get any better from that first stride. Or you can break it up into 5 sets of 20 canter strides interspersed with a nice working trot and pretty much guarantee yourself a markedly better canter by the last set, not to mention a nicer trot to go along with it. Every transition or half halt will help her rebalance and you'll find she stops falling on her forehand every few steps. She obviously has a bit more spunk when jumping, so kudos to you for recognizing the dressage plays heavily into jumping. If she gets strong and wants to start plowing through fences, half halts will be what gets her back.
You just have to see your distance...you don't have to like it.