I have had a history of dislocated shoulders, both complete and just in-out in a sec. My left shoulder had an operation years ago, but my right shoulder is still au natural. I have not been a good boy with regards to rehab excersizes and have always done just enough.
Now my right shoulder is acting up and I have got in-out’s several times lately at just normal everyday situations.
I have started the old rehab excersizes, strengthening rotator cuffs (easy stuff), and already feeling better. But this is what I always do, and this time I want to proceed and build my shoulders stronger. What strenghtening and stabilizing excersizes for shoulders, back and chest are recommended for people with rotator cuff issues?
I have access to weight room at work and rubber bands at home and during travels.
I use this to build up my shoulders.
hope this helps with your goal.
I have a rotator cuff injury, like you I did all the usual band stuff for a while to stabilise it. During my period of physio I kept going to the gym and listened to my body, and was echoed by my physio: If the movement/exercise aggravates the rotator in anyway then stop doing it or go much lighter until it is not aggravating any more.
I had to stop doing certain exercises all together for a while, mainly overhead stuff like presses, I had to significantly reduce other things like front and lateral raises. But as I got stronger and the injury healed I could keep increasing.
Same for chest exercises, just listen to your shoulder, unfortunately you use the shoulder for most things.
I should still do the band exercises but I don’t (naughty) and I still get issues if I sleep funny etc. But for the most part its not a big issue anymore 
I’ve had surgery on one shoulder and broke the collar bone on the other side that later required surgery too, both required a fair amount of rehab.
The best exercise that worked for me was lying, hips on a yoga ball, arms in push up position so you are level, then lift the non-injured arm and balance just using the arm that needs the work. You’ll need to shift your arm more central but the instability of the ball keeps you moving and brings all the muscles into play.
After the first surgery and several weeks of physio, exercises etc I still couldn’t lift my arm above my head, one week on the yoga ball and I had almost full range of motion back.
I had bruised the ball joint of my shoulder about 15 years ago, and again separated the AC joint 2 years ago going over my bars during a race. The best strength exercises for me have been shoulder fly’s and back fly’s. If I don’t do them for a while, I have to start off for a week or so with less than 10 lbs. otherwise the shoulder catches and pops. Once I do the workouts and keep doing them, the shoulder pain does not come back until I stop doing the workouts again