I don't see how your experience is different from many of those on this discussion forum, and many people who feel they have trouble making eye contact. I don't say this to be rude or to downplay the severity of your experience. By the number of people posting on issues such as eye contact or other things like shaking or blushing, this is a testament to just how much this affects everyone, especially here - people with social anxiety.
I say this also, the fact that your problem is not unique to you, in hopes that you find some solace in that others have had the same situation and that they have made progress out of it. I know plenty of people who tell me about their physical symptoms, including eye contact issues, and these things do impact their jobs and life because these things become very real. Yes, other people will notice if you avoid eye contact. That's the social world we live in. We all know when someone is not comfortable performing the behaviors which we have come to assume as a group to be normal and expected in a certain society.
You have tried CBT. That's great. How much, how long, did you apply it? Therapy doesn't come down to one or two or three ideas that we practice and then it all gets easier on a straight line heading to decreased anxiety. Over time, the accumulation of it all, yes, it can. But, in truth, it really does take time. It takes time and then it takes repetition and then it takes...wait for it....MORE REPETITION. Sigh... That's not the answer that any one of us want to hear. It's reality.
When we do the therapy, I also have to ask - are you applying the therapy. I have spoken to people who have done the therapy for years, and it's not working. Naturally by the time I speak to them, they are ready to give up. For me to tell them to do it again, this time with a different perspective - it's not exactly what they want to hear. But usually, after I've had a chance to listen to them, to see what they are doing, and what they are not doing, we can find reasons that the therapy isn't working for that person. Perhaps the person is really resisting the therapy. Perhaps the person isn't even applying the therapy. Perhaps the person has done it all and understands it but isn't even really able to identify ANTs and does nothing to practice not accepting them. That's tough. Now I have to say, let's do it again, and this time, let's actually apply what it says. And perhaps that person needs time to let these ideas sink in. The therapy really is about all of life. Some people catch on and accept pretty quickly. Others have real blockages and need time to accept and move past. Others have been doing it alone, without any opportunity ever to bounce ideas off of anyone else. When such a person finally bounces ideas off of me, and when we talk for a bit, it's like a new perspective is open for them. And I haven't said anything that is not already in the therapy. I'm not doing anything special. But because they've had a chance to talk about it and see it in a new way, they're able to go on.
Your eye contact issues are social anxiety issues. Through the therapy, all the things you learn there will apply to everything. There's not one exercise that dispels eye contact issues. It's the whole new belief system you are buidling.
In the last local group, I worked with an individual who had done the therapy, perhaps many times over. Still it took us going out together and working on eye contact exercises for this stuff to really start sinking in for that person.