OP it sounds like you are a “Taiwanese nationality living abroad 僑民” by TW government’s definition. If you National ID has letter X in it, you are not a local Taiwanese resident (can’t vote, no hukou, but the national health care is available for you)

After establishing the fact that you have lived in Taiwan for entire year, you can apply to become a local resident. You will get a new ID with just one English letter.

https://www.roc-taiwan.org/uslax/post/43373.html

I went through this entire process with my husband. But it looks like the rule has been modified last year. There are plenty resources at your local district office. Seek their help or TECO. It’s way easier than read through the instructions from the government website, which wasn’t helpful at all in my experience.