I am double jointed in all my fingers, toes, shoulders, and have well above average flexibility generally . . . but don't believe it would qualify as hypermobility. That said, I noticed that my level of injury when competing in sports (soccer, boxing, wrestling) went way down after starting some regular strength training.
Would it be possible for you to do some carefully supervised strength training? Learning to move your body in normal/safe patterns, minimizing overstretching, and strengthening your connective tissues would help protect you (especially joints) from injury. A proper weight lifting routine would be able to meet these goals.
Whatever sport you do, be very careful to pay attention to form - especially during the learning phase. You need to drill motions so that they become second nature in any sport . . . but in your case it's super important to pay attention early on to catch any unnatural over-extensions early and break the habit. It will be more important for you than others that you very gradually increase exertion. Overdoing things too quickly and getting into a very exhausted state will break your form. You need to progress, but very small steps over a long period of time will be less risky.