Just a quick update that I forgot I could use my phone's 3G internet connection on my computer, so I switched to that and the problem is gone (obviously it's still a bit laggy but at least I can actually play.) Still, that means that the problem is tied to my main house internet connection somehow, despite the fact that someone else in the house is using that exact same connection and is playing fine.

I'm totally stumped on this one, but just because I can play now doesn't mean I'm giving up, either! I'd like to use my main internet connection since it's so much faster, so I'm going to keep bumping this thread whenever I can and look for more solutions.