Alright, I finally got sick of the crap and went to support to report my issues. Turns out Weebly doesn't play ball with a full browser cache, and it causes Weebly to have all sorts of big problems.
Clearing my cache seems to have solved all of my troubles. I'm saying this for the sake of people who are tempted to sign up for a Weebly account, or who already have one. If Weebly starts malfunctioning and behaving like it's riddled with bugs, it might be the cache. Clearly it and restarting the browser will probably fix it.
Which is good, because I was also having trouble getting in contact with the support team, and was literally in the middle of writing up my migration notice when I was finally able to get through to someone on the team. There was a weird form submission issue, and I couldn't use it. And the chat box wasn't responding (which may have been connectivity issues on my end.) So the title of this exact post was very nearly "Abandoning Weebly" with an explanation of my troubles followed by an announcement of my plans to start looking for a different blog service.
Thankfully that seems to have resolved absolutely all the issues I was having, because I really do like Weebly. It's a bit on the advanced side, and probably would overwhelm beginners, but it's really good otherwise. (Sure as fuck beats Tumblr, that's for damn sure.) But it's not a community-driven system (which might be for the better) I can't subscribe to other people's blogs, favorite, reblog, etc. But at the same time, if I offend the wrong people, they can't mass report me, or reblog my posts to get more of their squealing, crowing, bigoteer sheep to dogpile me.
Like that word, by the way? I've decided to stop calling them Social Justice Warriors. The term "Justice Warrior" is a little too good for them. So I'm calling them bigoteers, now. It's "bigot" with the "eers" suffix. IE: Engineer, musketeer, auctioneer, profiteer, etc. It's harder for them to wear that as a badge of honor.
Example: "I take pride in being a warrior of justice!" versus "I take pride in being a bigot!" Yeah, not going to go over well with the rest of the world.
Anyway, don't worry, I still have most of the blog post that got ruined. In a moment of sudden brilliance, saved a large portion of it to a word document, knowing Weebly would screw it up. I managed to save half. I just have to muster the fortitude to rewrite what I lost.