I'm not sure about the extent of your shoulder with dislocations from what you say you at least know your limits. I am curious though is it pain you experience or concern about another dislocation?
As mentioned a crossbow will be a good choice, especially for us here in NC, but find something you can load properly. Will also recommend the 243, good round with minimal recoil for the most part. You may also be able to find a shooting rest or swivable shooting bench.