It sounds like you have Neuropathic Pain. My husband had 2 rotater cuff surgeries and he has pain down his arm to his elbow. He said it feels like he hit the funny bone, but it's constant.
When was your surgery? Are you sleeping in your bed? Prior to my rotator cuff surgery, my ortho told me to go home and "make friends" with the recliner and start sleeping in it...and because I will need to sleep in it for a couple of months AFTER the surgery. If you do this, it may help the pain; it's important to keep your arm vertical until it heals. Hope this helps you. I had almost no paid post-op; I took maybe 3-4 pain pills total after I went home.