1 Arm Everything! Rows, pulldowns, shrugs, overhead press, bench press, incline. 1 arm dumbbell push press. All angles! Start with the left side. Do your all your sets/reps, then match it with the right.
Sidebends. 1 Arm Farmer's Walk. Walking with the dumbbell locked out overhead. Lunges.
Tons of upper back and rotator cuff stability work. Scarecrows, cuban press with dumbbells, seated dumbbell cleans, shrugs with dumbbells.
Right now, you left and right side don't work in symmetry, so all bilateral moves are going to make things worse. You've got to make a conscious effort to get yourself lined up straight. Then, when you do your one arm lifts, you've got to force your body to stay lined up "square" and symmetrical.
Find the least challenging moves, the ones that you can do while holding your position. Maybe even machines at first.
So a progression could go like...
Seated Machine Press, 1 Arm
Seated Dumbbell Press, 1 Arm
Half Kneeling 1 Arm Press
Standing 1 Arm Press
1 Arm Push Press
Here's an awesome shoulder warm up from Diesel Crew
Blast straps, rings, or TRX straps are great too.
How are your hips and feet? Are they even, or out of whack also.