I had surgery several years ago on the shoulder of my bow arm. I had a torn labrum, but fortunately, the rotator cuff was o.k. Also, I had a torn tendon in my chest muscle. After the surgery, I had several weeks of therapy. I recovered most of my strength and flexibility. I can still shoot my hunting bow (53#) just fine, although I did have to reduce the draw weight down to 40# for a while. The only problem I still have is that I can't pull the bow when my bow is canted to a totally horizontal position. This has hurt me on a few funky 3D targets but has never come up in a hunting situation.
Discuss your situation with the doctor and let him know exactly how it feels. He will want to know what you hope to realize from surgery or whatever treatment is used. I let him know that my archery was my primary concern, and he addressed that in my therapy routine. Sometimes, injections and therapy are all that is needed to solve the problem. It is possible that pain will be projected down into your arm. That happened to me. Whatever you do, discuss this fully with your doctor and follow any instructions given. This also applies to the therapist. Don't try to ignore and power through the pain, as it won't go away on its own. This situation is bearable and fixable, so hang in there.
Now, my other shoulder is having issues...