Does this also happen with a normal "Save"? Can you perform a "Save As" before you add your annotations?
Acrobat performs different operations when you select Save vs. Save As: With a normal Save, Acrobat will create an 'incremental update", which means that the original file will be left as it is, and any new or updated information will be appended after the end of the original file. When you do a "Save As", things are different: Acrobat will recreate the whole PDF structure from scratch, which of course takes much longer.
If there is a problem in the original PDF file (e.g. a corrupt PDF structure), a Save may not even know about that problem (e.g. because you never displayed the page with the corrupt data), but a Save As, because it needs to touch every bit of information in the PDF file will come across the corrupt portion. Usually, you will end up with an error message, but it's possible that you are triggering a bug in Acrobat, which can cause the crash.
It still crashes when I save before editing the PDF. But i noticed it seems to only do that with one document (the most important one, of course). What Can I do to fix it?
That's what I suspected. Your annotations have nothing to do with the crash, the document is very likely corrupt. Unfortunately, there really is not much you can do. One option would be to use a different version of Acrobat/Reader to see if this is something that only happens with your version of Reader. You can still download Reader XI from here: Adobe Acrobat Reader DC Install for all versions