Several hours more… I finally succumbed to switching themes and plugins.
Turned everything off - Default theme, no plugins, no htaccess… And it worked fine.
Turns out that the plugin “Lightswipe Masonry” (Some gallery thing) caused a conflict with galleries or something when saving posts - Because why would it…
Didn’t even look at the code to figure it out - Immediately deleted it after confirming the issue came back when the plugin was active and now the site seems fine.
Wtf!