When a speed horse goes bad, the best thing for it is to forget about speed. :) If there is a lack of absolute control at a walk, that lack of control will be amplified at a trot. I think you are starting with two good, achievable goals. Walk, bend and flex at a walk in both directions, learning leg and seat aids at a walk, walking concentric circles in both directions, then transition work. When you can do all that at an even jog/trot without breaking pace, you're ready for new goals.
I recently had to work a horse out of a tendency to engage a strung-out, breakneck trot when asked to move out of a walk. There was no collection, no rate, no evenness...just walk, then head-in-the-air wiggling trot with lots of spooks thrown in for good measure, because her last rider would then walk her back to the barn on a tight rein and get off. Circles every time she increased speed on her own helped, but not as much as the walk/trot transition work. I'd have her trot ten or twenty strides and then sit down and ask for the walk. We'd walk ten to twenty strides, and I'd pick up the reins, shift weight, and ask for the trot. Eventually, she stopped anticipating that the trot was going to get her anywhere, and she stopped trying to get "there" so quickly. The trot became smoother, she began looking for the cue to slow down (because she was anticipating it), and she became very responsive to weight and seat and voice. After that, circles were used to check any increase in speed. Last time I rode her, we were doing a lovely western pleasure jog on a loose rein, with only the occasional need to check her speed using circles. Also? Those spooks became a lot less frequent. Good luck.