My app just went semi-viral and it's slowing to a crawl!

You not (but all user yes). But other app in the same cluster yes.
When you are on personnal or hobby plan you have a base server capacity shared with some others app. (For example, 4 apps). So I guess that Bubble have something to prevent an app to use 90% of all the server capacity and will limit it to 25% (fictive number, but Bubble team may answer this).
Now, when you go on a Pro plan or more, you Get reserved capacity server. So instead of sharing it with other app, you have your own capacity and you will be able to reach 100% of it.

What you experience is exactly what is reported in thi topic: