Somehow I managed to solve this issue, but unfortunately I'm not sure what did it.
I started to fiddle around with anything I thought could have something to do with the problem, and suddenly it started working again.
Here's what I did.
* I removed Sublime entirely using AppCleaner, including all support-files. I have my settings symlinked from a git repo so restoring was not a problem. However the problem persisted.
* I tried to do this again, but this time removing everything in my User-directory except .sublime-settings-files (eg ca-bundle, caches, etc).
* I uninstalled my other python3 installation. I had it installed with homebrew, so I did `brew uninstall python3`.
This one is my best bet on what did it. Since I was fairly sure that the problem had to do with a filewatcher, and the problem persisted even after a complete reinstall, I thought the problem might be with Python. Sublime 3 comes bundled with its own Python3-interpreter, so maybe it clashes with the homebrew one somehow. I have nothing to prove that this is the case, but I can't think of anything else that might have solved the problem.
Try these if they apply to you, and if anyone have an explanation to what's actually happening I would appreciate it!