Same issue here too, only it's been happening for many months. If I open the note I can type exactly 1 thing and it will save, (technically say "Edited"), but the second I pause, even to go and correct a typo, it immediately says "Saving..." and never saves. It's incredibly frustrating as I do all my meal planning and grocery lists in Notes. I type it all up on the browser version and access it on my phone at the grocery, and my iPad when cooking. It has screwed me over at the grocery so many times.
I have attempted using a different browser and that doesn't fix it. I have attempted creating entirely new notes (instead of using the same one and updating it), and that doesn't fix it. It is beyond ridiculous from Apple that something this simple continues to be broken. I live or die by my notes, as I always say, and for them to repeatedly fail this spectacularly is incredibly frustrating.