“I have to go sleep” seems like it should be accepted.

is it possible to say: "ich muss zu schlafen gehen"?

My ESL students often ask about the difference between "I will do this" and "I am going to do this." Both are future tense. I was interested to learn that "going to do" is a fairly recent development in English. Shakespeare used "going to + verb" only when someone was literally going somewhere.

To me, there's something similar with "go to sleep" - which often means "fall asleep." I don't think this is the case in German. "Schlafen gehen" seems to mean leaving one place and then sleeping in another.

https://learngerman.dw.com/en/gehen-infinitive/gr-38308248