You obviously put too much water. There should not be enough to leave drops, thus "dampen" is the idea.
Mistake number 2 (and the biggest one): You should
never wipe the glass with anything other than the microfiber cloth. That is how you end up with streaks and, depending on the material of what you are using to wipe, even scratches on the glass surface.
You should have dried the microfiber cloth sufficiently so that it could be used for the drying.
I apply a very small amount of water to one corner of the Apple cloth and then gently wipe the screen with it. Then I immediately clean the screen with the dry parts.
I have one but I don't think you are going to like it.
It sounds to me like your shirt has scratched the glass panel. I was going to suggest you repeat the cleaning process as I outlined above and that should have done the job. That the "smudges" persist is not heartening.
Maybe you should take it by an Apple Store. Even if you crack a 24" glass panel they aren't as expensive as one might think. Someone in another thread who had his glass panel replaced recently in a London Apple Store mentioned that it only cost him $36:
https://forums.macrumors.com/posts/4893193/
If it is just smudges and not anything physically wrong with the panel you should be able to clean it with just water and the microfiber. You didn't use anything other than water, right?? No Windex, etc