To anyone still following this, I found another potential solution that does not require using Chrome if you are already on Mojave.
The Safari Technology Preview can be installed without disturbing your current installation of Safari or MacOS:
https://developer.apple.com/safari/technology-preview/
The current preview version 97 does not have the cross-site cookie bug (the Prevent Cross Tracking option works as intended) and also does not have that annoying Safari 12.1 behavior of auto-submitting login forms when you use user/password fill.
The downside is that it keeps its own set of bookmarks and history so you might need to do a little work to copy over your bookmarks.
If you go to the Safari Technology Preview download page, DO NOT repeat DO NOT download Safari Beta versions. If they don't work you are basically screwed. The Safari Technology Preview can be simply dragged to the Trash if you decide not to use it.
I am hoping that since the preview doesn't have the cookie bug that Safari 13.0.5 will fix it as well.