I completely agree with taking the draw reins off. They get in the way of teaching a horse to properly engage and use their back. I don't like them. I used them in Germany because I didn't have a choice but I have not used them since. I think it is very counter-productive and covers up the real problems. He's not learning how to work into a contact, balance himself, or work in a consistent rhythm. All components which are drastically more important than where his head is, regardless of above or behind both are evasions to contact and using themselves properly. When you have the timing and consistency for those 3 components he will reach into the bridle and connect. But he won't learn to work over his back just by his head being down.
Big problem is this horse has no consistent rhythm, his balance is all over the place and he is WAY too forward. He goes through his outside shoulder quite a lot, bending him more to the inside will only make him fall out more. I think your stirrups are too long which is part of the balance problem and him being so off balance makes it that much harder for you to balance. You also need to breathe, big and deep breaths and calm your body and energy. Post slower and count 1-2, 1-2, 1-2 for trot and set a consistent pace. That makes a BIG difference if he is too quick (which he is), I would not have chased him into a racing trot. It also helps to half halt in the sit phase by tightening your core and if there is no reaction, tighten your thigh for a moment and then ask with your core and thigh, as well as hand and expect a reaction. And if you're struggling at any point just transition to walk. Sometimes doing more is just more and is not as productive as doing less but quality work.
For the leg yielding which is a VERY helpful exercise you need to slow him down. He needs to learn to respect your half halt and not barrel through his outside shoulder. The purpose of the exercise is to help teach him how to use his body, keep his shoulder in line and learn to yield to the leg and respect the outside rein so he can be more connected and through in his body.
What I would do is go to walk and practice turns on the forehand. Both directions and work on transitions walk-trot walk-trot and make your aids clear. He needs basics. Turns on the forehand, leg yielding, trot a few steps transition into walk and back into trot and play with how little you can do to get a reaction. Start with being as light as you possibly can be and expect a reaction, if there is no reaction gradually do what you need to get a reaction but don't expect instant, a correct reaction is more important than a reaction. So it may take a few steps. Try to find a way to do more with less. You don't have to use so much or so many aids. For example I ride a horse (not my personal horse) who is just broke but he still can transition downward or halt from me literally just closing my thighs slightly, exhaling, half halting in my core and pushing the reins forward. I always give (my rein doesn't get longer but I put my hands forward) in a downward and upward transition to encourage and test self carriage. If the horse comes above or behind (behind is too light in the contact), I do the transition over because I got the timing wrong. And if I have to hold a horse into a transition something is wrong either in my training or timing of aids.
Being behind the vertical, I think a lot of babies go through that stage when they're figuring out contact and learning how to organize their bodies, some do it when they start getting tired because it is easier than carrying themselves . It took a lot of time to train my horse how to accept contact and use his body correctly but it was worth the effort. It took a lot of laterals, transitions, and correcting him when he'd get too light in the bridle. (I'd keep my contact and add leg) and at halt add leg so he understood it meant to push his poll out. Anytime I felt he was too light I'd move into laterals and if he blew through his outside shoulder I'd go back to turns on the forehand and emphasize he move his haunches and is not allowed to plow through his outside shoulder. I did this by closing my outside knee and thigh and bumped him with my outside leg in the leg yield when he felt like he was falling through his shoulder. And if necessary you can tap a horse on the outside shoulder with the whip (tap, not smack). As well as half halting to keep a slower tempo and make him use his back.
Your hands are too high for this stage of training, I would lower the hands, relax the elbows and lower the outside rein to his wither to help prevent him from going through the outside shoulder. Think of the outside rein as the steering, balance, and connection rein. Inside rein is for suppleness, inside leg is for bend. Anytime you feel yourself pulling back against him, I'd not and either shorten your rein so you feel a connection and encourage him to stretch to the rein. Organizing a baby horse is NOT easy, most of my riding experience has been developing young horses, green horses or re training horses. I had one who showed 4th level and knew the movements but had no idea how to carry herself or connect. She was ridden walk-trot for a while. It took A LOT to one keep her from running away and two to get her to trust a rider's hand and trust a rider enough to let the rider ride her and not just force her into a frame and make her do something.
I say all this not to be nasty or hyper critical but to say here are some things to pay attention to. Also believe in yourself and be patient, good things come from patience and time. I go into detail because riding is that detail oriented and that's how I process when I ride.