Mame, I had almost exactly the same condition about four years ago - a full tear, mostly the result of accumulated wear. I noticed it during workouts when I rather suddenly could no longer do straight presses or pull-ups without severe, sharp pain in my right shoulder.
I had the rotator cuff repaired (stitched together) at Johns Hopkins and the repair was successful. Thr recuperaqtion and physical therapy took about eight months during which I gradually restored my range of motion and the strength of the associated muscles which atrophy very quickly after the procedure. Now , four years later I am still pain free and have resumed my former workout activities without reduction.
My best advice would be to get the repair and put a lot of effort into the physical therapy. Let them know you want a full recovery of range of motion and activity and keep at it until you get there.