Ouch! I had a torn labrum (with a few other things thrown in for fun), but didn't have the torn rotator cuff. How bad is the labrum tear?
I did a cortisone shot and then had surgery. Not every tear needs surgery, but mine was bad enough to need it. The cortisone shot helped while I waited for surgery. I have full use back (again, different injury so your mileage my vary).
Shoulder injuries are awful. Recovery from shoulder surgery is hellish (that can't be emphasized enough), and your shoulder probably won't ever be 100% the same again. But talk to your doc about all your options, and make an informed decision based on your particular injuries. I don't regret my surgery now, but if you asked me within the 6 months following my surgery, I might have.
Oh my gosh, rotator cuff injuries! Worst and things people really struggle with often. I take it you are seeing an orthopedist? This would be best if you are not. I'm all in favor of least invasive first. You've tried PT. But the degree of tear may mean that it is going to need more. Sigh. Yes, try the cortisone. I'd do that before surgery. Problem with surgery is that sometimes patients don't get the full range of motion back. But if you are in a position of needing it, you need it. Rotator cuff injuries cause us to not be able to just get on with life ad do what we want to do which is frustrating. Try the cortisone but I imagine they will begin to talk about surgery for you.