Sounds like this is classic
Cubital Tunnel Syndrome. Near the bottom of this page has some non-surgical treatments like taking anti-inflammatory medications (NSAIDs like Ibuprofen, Naproxen, Aspirin, Diclofenac, Meloxicam, Ketorolac, etc.. Some of these you can only get by prescript
ion.) Like your doctor said, you can wear a brace. You might go to a medical supply store and explain the situation to see what they might recommend...a pharmacist might be able to help with this too. You can google medical supply stores in your area or try your local yellow pages to find one near you. There are also the nerve gliding exercises that can help and I'm sure there is a lot of information on the internet about
which exercises would be best.
Since you have pain in your neck and shoulder too, you might want your doctor to make sure there isn't an issue with the ulnar nerve in your neck or shoulder before jumping into the elbow surgery.
I've had problems with the ulnar nerve in both arms but very minor. I had the nerve conduction tests done and they were negative. The doctor that did the tests said that most likely I was putting pressure on the nerves by resting my elbows on hard surfaces which caused slight pinching of the ulnar nerves. I just avoid resting my elbows on anything now and I don't get the numbness or "pins and needles" sensations anymore unless I rest my elbows on something by accident. The doctor also suggested to try not sleeping with my arm bent at the elbow as that can cause these symptoms as well. There is no surgery I can have done, I just have to be careful not to rest my elbows on anything and not sleep with my arms bent at the elbow.
I did have carpal tunnel surgery on both wrists so I do know how these issues can be very painful and debilitating.
From what you described it really sounds like you need to have the surgery for the ulnar nerve to be repositioned in your elbow so that there is no entrapment/compression. Elbow braces are only going to help a little and are not going to prevent the numbness and other sensations entirely.
Don't wait too long for surgery though because this can cause permanent nerve damage and irreversible muscle wasting/atrophy. I'm not trying to scare you but if your doctor says you need surgery then you should probably go ahead and get it done or things will just get worse.
EDIT: Here is a video of the nerve gliding exercises:
Nerve Gliding Exercises for Cubital Tunnel Syndrome