Train Your Back. Teach it what to do, learn how to control it.
Make sure you know how to move your body around, using your back. Go to the Smith Machine, and put the bar all the way down. Do pushups, squeezing the bar as hard as you can. Feel your back supporting you, and controlling your elbow position. Spent some time at the "top" and "bottom" position. Think of it like a moving plank. Get tight.
Then set the bar to waist height and do inverted rows. Use your back to pull your chest up to the bar. Squeeze and hold it at the top. Again, get tight. Learn how your back muscles control where your arms will go. Feel the tension. Pull to your chest, like the reverse of your bench press. Pull to your clavicles for more upper back. Drive your feet into the ground. Get something out of it.
Then do shrugs, or rows, or some kinds of pulldowns, whatever you can do, so you can do some reps left, some reps right, some reps together. Like Iso pulldowns, 8 left arm, 8 right arm, 8 together. Make it feel the same.
Seated dumbbell shruggs. 10 right, 10 left, 10 together.
Also, do a bunch of shrugs with a barbell. Every few sessions, change the angle. Like leaning slightly forward, or laying on a bench, or standing with a snatch grip. Or with a trap bar. Squeeze and hold. Feel it, make it feel the same on both sides.