Some things to think about.
As already stated outside rein support both directions and organization. His tempo is too quick and it's affecting his balance, so gaining the strength/coordination will improve that a lot. Half transitional half halts help a lot with organization in the canter, so they learn to respect your half halt and slowing his tempo without causing a transition. Being too quick throws him more off balance.
Going left he wants to be crooked, going right he doesn't want to bend through his rib cage. Both directions he falls out through the outside shoulder and leans in. This is a fairly normal crookedness but something to be mindful of.
What I always focus on when riding is are they through? Are they seeking the contact? Are they independent of my rein? Can I put both reins forward and they still seek the contact? Can I pet them with my inside rein and turn? If I ask for a half transitional half halt, do they listen? If I open my hip to send them forward, do they listen? Can I ask for a little bigger gait or a little smaller gait and they respond? Or I kinda have a checklist, for me I prioritize throughness because when a horse is through, they're relaxed, in balanced, with good swing in their back and they're organized.
Other things that help. Shoulder control, controls the balance and organization. For example if I'm riding a very hot, nervous or spooky horse or a horse who doesn't focus well. I do more shoulder in and try to get the attention through exercises and giving a clear focus. I don't usually drill or do the same thing over and over again. I might do something 3-4 times and move onto something else and come back. I don't expect perfect but better when I ask for something. I don't think of it as riding a moment but training a reaction. Giving a horse something to focus on, rather than their anxiety or nervousness or whatever is distracting them. It's not always easy in the beginning but it gets easier as you condition the response to focus and listen.
Half transitions, ask like you're going to transition downward then send forward. That's usually how we introduce a half halt stemming from core and seat and actually get the "sit" and increasing engagement vs the dropping out of the transition. It's making the more subtle aids clearly understood without confusion. This half halt is very helpful for preparing for changes of direction, shoulder in/fore, as well as for shoulder control and improving laterals.
Consider counter bend. I don't say flexion because I do counter bend more from my position and legs asking for the bend vs my reins but I will half halt on the inside rein in counter bend to maintain the balance. Flexion I think of as to soften the jaw and bend I think of as the rib cage and shoulders. My reins are to supple the jaw, ask for more flexion or half halt or to allow the bend to change (if you have a tight outside rein and not allowing the bend, you're not going to get the bend through the inside rib cage when you switch to true bend) but you still need to maintain the outside connection which is the part people usually lose when returning to true bend. The issue here is not enough connection to the outside rein which counter bend can help with, as well as leg yielding off the outside aids. I also aim for one 12m circle off my outside aids (once warmed up) when I feel a horse go through my outside aids, if they ignore my correction, so that my expectation is clear to them but I still support the balance with my inside leg so they don't just fall out through the opposite shoulder.
Shoulder in/fore into transitions (more at walk/trot and prepare for canter depart) and while on a circle also helps establish the inside leg, outside rein connection and to establish that communication. Part of why I school shoulder in is because I want an effortless reaction from my position, so I hopefully don't have to leg yield off my outside aid or I have something I can go to when I have a nervous, spooky, or distracted horse. Shoulder in comes from the shoulder coming in with adequate bend in the rib cage and shoulders. Biggest problem I see in shoulder in is people lose the throughness because the horse sucks back in it's neck in a lateral because they lost the impulsion. It's good way to help with setting up and training for good transitions. They should step into it. It also goes back to at the walk and trot and being able to pet the horse with the inside rein and still be able to steer, get shoulder in or fore without the inside rein and off the outside rein/knee while supporting bend with your position and inside leg. Or like you're maintaining the same bend from a 10m circle down a line or if you're on a circle exaggerating the bend coming from the shoulders and supported by the inside leg/position.
Another is have a connection on the outside rein, maintain your circle and pet with the inside rein. Or go down quarter line and aim for a straight line. If getting very crooked, send for ward first before making corrections.
Another exercise is spiral circles leg yielding in off the outside leg to about 12-15m and leg yielding back out to a 15 or going down the long side leg yielding off the track to quarter line for straightness. Lateral work isn't to complete a movement but they're good gymnastic exercises to improve the basic gaits, balance, rhythm and general organization of aids and make the communication clear between horse and rider.
For you core stability and strength will help a lot with just about everything and being mindful of your shoulders. Strengthening them will help. Shoulder position does affect hip position and core stability.
Overall you're doing well together and you should be pleased. Dressage is something of a jig saw puzzle, lot and lots of pieces that all fit together. Always something more to think about and work on.