You need to keep your hands quieter...especially during the canter I noticed you tend to "pump" your hands. Maybe work on just keeping them "planted" by his withers for now so they stay more still. Also, work on keeping momentum. You should have the same rhythm throughout the circle. Every time you get to the line before the fence he slows down and loses momentum and then lunges forward to rush the fence, and you let him. I would do more circles before the jump and not let him just go straight over every time.
He's cute :) Reminds me of a tb I was riding earlier in the year.
Also, work on getting him moving out and more forward (the trot at the beginning of the second video is very up and down.) He's a bit heavy on the forehand. Rounder movements will help your momentum and timing, as well as help him jump rounder (preventing the rushing will help, too.) Gymnastics are a good tool for this. You'll have to be sure you're more organized between jumps, though--shorten your reins a bit so you're not dropping them so far down to slow him after the jumps.
Madison, I noticed a lot of head-bobbing, especially at the beginning, but I don't know if I saw any lameness in the legs? She doesn't have the quietest hands, it could just be the contact. He seemed to want to jump, and if he was ouchy I think there would have been more resistance.