Carpel Tunnel usually happens because of a repetitive motion in an axial direction. Machinists get it a lot when they manually move the table with their wrist revolving it instead of cranking it, they do it with their fingertips. It is much quicker to slide the table across, but you are beating the heck out of your wrist. I cranked it to avoid this problem. I have known many people with this problem. Usually they open up the space where the Median Nerve passes through. They may relieve the compression by cutting some bone, or some ligament, or both. It depends on how much space is in there as to whether they go doing arthroscopy, or cut a 2" slit. The prior method heals much quicker, but I've know people who ended up having to get the surgery again because they just couldn't get a good enough look inside there, and didn't remove enough bone or ligament.
I imagine your feeling tingling in your thumb and fingers. Here's the problem, the longer you wait, the more scarring on the Myelin sheath of the nerve can happen on and once the pressure is relieved, you may end up still having the tingling, but it won't get worse. Not getting it will mean that you will lose strength in that hand. This surgery is done a lot and has a very high success rate, the problem is if you wait too long. Nerves don't heal. If you catch it early, your hand could be back to where it was, or almost, and you will be better off for it.
There really is no non surgical method because the damage has been done. Some people try wearing wrist braces to limit movement, and get the Nerve swelling to subside, but the second they start doing stuff, it comes back. I'd get a second opinion from an aggressive surgeon, and personally, as one who has had 5 major Spinal Cord injuries, get it fixed as soon as possible. When I first broke my neck, Cat Scans had not been invented, and they used to argue whether a ruptured disc was cartilage or a tumor, because there was no way of imaging. By waiting too long, I ended up with permanent damage to my neck, with scar tissue forming of the superficial nerve, median nerve, and the rest of them, down my arms and my hands. I really never recovered like I would of of if I hadn't had such a chicken sh*t conservative doctor. I was young and didn't know better.
In a nutshell, the median nerve is compressed, the pain is traveling, not good, the mechanical damage to your hand will increase, as the damage to the nerves outer layer (Myelin Sheath), it's insulation, gets scarred. Worse case, it could start sticking to the bone, and then you are in for a life long burden.
I would get the surgery, nip it in the bud. That is the best case scenario. Inflammatory medicines will not work because the problem is mechanical. My brother in law waited for years, now he can't ride any of the motorcycles he has. He has something like 10 of them. I've known many machinists who developed this problem, the ones who got the surgery really never talked about it again, the ones who waited ended up not being able to operate the machinery. They had trouble driving, turning door knobs, etc.. It sounds like you talked it over with your doctor, there isnot magic pill or exercise than can fix it. The opening is too small, the space has to be opened up so everyting going through there moves easily. It's a very mechanical problem. When that nerve swells up, it will only get worse, and there is no pain relief on the market that will make it feel better. Everytime you feel a tingle, that is the insulation on the nerve rubbing. Unfortunately, the only the way nerve can fix that is with scar tissue. That scar tissue causes lingering pain. IMHO.
