The one common denominator in all shoulder rehab is that you are going to have to exercise your shoulder it at some point to get it working properly again. Whether this is instead of an operation or after an operation can depend on how you treat your shoulder in the meantime.
I managed to tear my rotator cuff lifting something that was too heavy resulting in too much force being put on the tendons of my left shoulder. They tore. I know that because I heard them go. Just a quite pop but I knew that I had done something bad. This is called an acute tear. It is the result of a sudden force. You can also get acute tears as the result of a fall or knock but the most common cause is sports, specifically sports where you are throwing which is why rotator cuff syndrome is sometimes called pitchers shoulder.
With an acute tear you need to immobilise the shoulder for a while to let it heal before you start any king of shoulder rehab regime. Using a sling is a good idea but definitely avoid any action that causes pain or you will just make the injury worse and end up needing surgery. You also need to treat the inflammation using anti-inflammatory drugs and ice packs. If the pain persists for more than two or three days see your doctor who may administer a steroid injection to reduce inflammation. Once the inflammation and pain have subsided you can begin exercises to strengthen the rotator cuff muscles. These will often be Yoga or Pilates based as they focus on restoring control and flexibility as well as strength which is vital to avoid future problems.
A chronic tear on the other hand is brought about through wear and tear. It usually occurs in the over forties as our posture changes but can also happen to anyone who repeatedly works above shoulder height. A Chronic tear is caused by tendons getting rubbed against bone in our shoulder. It tends to start gradually with you experiencing pain at night and the pain gradually worsens as the tendon gets more and more worn. The shoulder becomes gradually weaker and you will start to experience pain as you lift your arm. It is possible that a tendon can snap completely if this condition is ignored but this is fairly uncommon simply because people tend to seek help because of the discomfort.
With a chronic tear you will again need to focus on reducing inflammation and managing the pain. Trying to avoid painful movements will help but in order to rehabilitate the shoulder you need to address the underlying problem. If this is age related, exercises to improve posture will help. Again Pilates and Yoga based exercises are good as they focus on control and flexibility and also tend to help with core strength which can directly affect posture. Get your posture back to how it should be and healthy shoulders quickly follow.
My one main tip for getting your shoulder back to full working order is rest. Do not ignore this advice like I did which is why I ended up in pain for six months and facing surgery.
However you have managed to injure your rotator cuff carrying on regardless will make it worse. Resting your shoulder is difficult, especially if it is your dominant arm simply because you use your arms all the time and consequently are using your shoulder as well. Change the way in which you work to avoid any action that causes pain. Simple things like rearranging your desk at work can make a big difference as can giving up driving for a few weeks. Not convenient but it will definitely help in the long run. Only when the pain and inflammation has subsided is it safe to start shoulder rehab exercises.
And once your shoulder is better, keep up the exercises and keep your shoulders strong.