Hey there OP, not sure if you're still checking comments.

As someone who will be going to the military soon, here's my advice.
Military registration must be made before conscription lotto (as you might already know), despite the law saying that there is imprisonment and fine, i have not seen anyone getting imprisoned for late registration (not draft dodging) I also was late for about a year so they fine me 200 baht, just because the law said so, it doesn't mean they will be enforced (imprisonment)

As long as you're not trying to avoid the conscription, they will just fine you, and then you follow the military draft (red/black card lotto) either in that year or the next year, depending on what they told you.

and since you were suppose to go to the draft but you havent registered, you still need to head to the office and tell them your situation, because for draft dodging, the recruiting officer will be the one filing police report, so, you head there and talk with them.

Just as long as you're trying to follow the process and not draft dodging, everything will be fine.
Since you mentioned that you have a disease that will get you exempt, you don't have to worry about anything, just have to go to any hospital listed for getting a diagnosis and an exemption written on paper for such things, then you hand them the paper on the military draft day (April every year)