That is a cute horse. I remember you said that the video was taken on a hot day, so might account for him not always going really forward with some drive from behind.
Yeah, you need the D saddle and nix the martingale.
The one thing I saw that I didn't see anyone else comment on is that when you were trying to get him "on the bit", perhaps by fiddling with your fingers, he came down and behind the vertical. Instead of reaching for the bit, he was coming behind it a bit, and putting his head so much behind the vertical and downward puts more weight onto his shoulder/front legs, ' Heavy on the forehand.
You said he was often "go go go" at jumping and I am going to guess that he had his head up and maybe pulled back to almost creat a "ewe" neck shape. ( I don't mean permanently, just as a reaction to the pull on the rein and his brace against it and the excitement).
Anyway, I would start his dressage training by working with him reaching forward and down and NOT coming behind the vertical, but rather staying a tiny bit in front of it. Then, you ask him for forward!
The correct development of the neck/shoulder muscles is very inportant. Dressage horses , when they are just starting, spend a long time working in this frame and at the lower levels, training and intro, the manuals emphasize relaxation and being just in front of the vertical.
He has decent rythm and working on geting him to reach forward and down to the soft bit will bring out his relaxation more.
I didin't see a lot of off balance, (video wasn't so good) but what isthere seems to come from him resisting the contact.