In reply to bowlie:
You are asking for medical insights about a condition very specific to you, which means those of us who attempt answers don't know what we're talking about.
Having fully disqualified myself, here goes anyway.
First of all, dead hangs are a training technique. Climbers don't do them while climbing. So the most obvious answer is, don't do any dead hang training. Of course, all descriptions of dead-hang training include admonition to keep the shoulders engaged and not just hang off the ligaments, but if you are doing such training to exhaustion, the chance that you will tire and at some point lose that shoulder engagement is significant, and probably a lot worse for you than most of the rest of us.
Climbers do rest by hanging on a single straight arm. You will either have to forego this useful technique or learn to keep your shoulders engaged. I don't think this is a strength thing so much as a necessary automatic physical response. It would make more sense to hang comfortably one-handed from big holds in the gym with your feet on as well and work on immediately engaging your shoulders as an intuitive reaction to the position.
What is going to be dangerous for you is dynamic moves, when your body is in motion and you catch. If you aren't quite strong enough or if your feet pop off, you'll be suddenly left hanging on one arm, which is probably going to end badly for you. I'd strongly consider staying away from such motions.
There is another way climbers dislocate their shoulders that isn't by virtue of a planned or unplanned dead hang. Shoulder dislocations can occur when the climber's arm is extended laterally to the side at should level and the climber is pushing down very hard with the hand. This is probably going to be a concern for you as well.
As for training, I think any kind of dead-hanging (e.g. with less than body weight) is going to be detrimental, as all it can do is stretch those ligaments. What I think you want to do is strengthen the muscles that keep the shoulders engaged, which would involve...engaging motions, not dead-hanging. Pullups are probably a good idea, and working the actual engaging motion with pulleys or resistance bands might help too.