My suggestion, Ryu, is to try rehearsing, in your head, the things you want to say prior to an appointment. Make it as short and blunt as possible. "Hello, Dr. ____. I am here because I think I may have a mood disorder. My symptoms are...." One thing to keep in mind is that doctors are used to hearing this, they're used to people being uncomfortable, and there is no way that you are going to be their worst patient. The more you say it to yourself, the easier it'll be to say it to someone else. It's awkward the first time because you've never admitted it out loud to anyone else. The first step is the worst in that regard. After the first step, it will probably get easier.
What you're going to have to overcome is everything you've put into masking the disorder from the people around you in everyday life. It's not going to be easy to undo all that. If you've chosen to keep this to yourself, you've chosen to have two sides, and you'll have to play both sides. In counseling, you'll have to forget the person you are in front of others. You'll have to be honest. A psychologist will probably figure out when you're dodging the issue, but that takes time, and time is a valuable thing.
If you have someone close to you who can support you, then don't be afraid to use that support. However, admitting to having a mental disorder comes with tremendous loss of face. Most people don't understand what a mood disorder is, and most don't want to accept that emotions are influenced by chemical interactions. I suggest that you ignore what those people say. What is normal for neurotypical people is probably not normal for you. I wouldn't blame them for not understanding, but I wouldn't try to win their approval either. The best thing to do is to ignore them for the time being and work on yourself.
Another thing to think about is that you've already been doing all that you can to fix the problem yourself. If that were good enough, you wouldn't be considering professional help. If you had cancer, you would see a doctor, and I'm sure you would keep up with your treatments. Think of this as that. If it helps, try to distance the disorder from you yourself as a person. The disease is a disease. The treatment is for the disease, not for you as a person. There's nothing wrong with you. It's just the disorder that needs treatment.
Also, it may turn out that you've misdiagnosed yourself, and maybe the condition that you have is not what you thought it was. Be prepared for that. You're not a trained psychologist, so you couldn't have known. It could be that the condition is much less severe than you thought it was. That could only benefit you.
Lastly, try to commit yourself to making an appointment. Get a calendar, write it down, or program it into your phone with the alarm set to go off. It won't happen until you make it happen. This problem won't go away on its own. I had concentration problems that sound a lot like yours. I was kicked out of school because of bad grades. School is hard. With a severe disorder, it gets harder. I'd like to see you avoid making the same mistake I did.
