a_ahmed: You sure it's the rotator cuff it's self and nothing else?
Usually when pushing down, along side the body (as in dips), the elbows in and close to the body, less of the shoulder girdle is involved.That's why it's one of the elite triceps exercises. All the while keeping the body in a somewhat straight up & down position, rather than leaning forward If having the elbows out and away from the body & leaning forward (like most guys do who think their working the triceps to the max), more influence on the joint/shoulders. Though your case could be all together different. Checked medical opinion, I hope.
Would suggest BB overhead or lying/incline tricep extensions, but they could pose a problem. Again, not knowing the extent of the possible tear/rupture in the shoulder area. Though a DB could be better served, than a BB, if no pain is present. Like with anything else, with a DB, the hand position and angle of the movement becomes better adaptable to the path of that movement.....more flexible. While a BB is so much more fixed for the hand position/path of a exercise. Experienced BB'ers know this for exercises like benches/inclines, arms and just about anything else, when using a DB over a BB..
Might suggest one arm tricep press downs (lat/cable device). Some have built up to impressive weight and size with these. Kick Backs/Triceps push off's is another superior exercise, allowing a full triceps stretch. You can adjust the focus/resistance of this exercise, from start to finish.
Some have trained around injuries for years, which is possible though not the best idea. But you will never capture the full benefits of a good and productive workout, with the body not being 100% functional. Good Luck.