Everything is all patched up finally. The site and the API are both up, running, and seem to be handling traffic as well as ever. Thanks to everyone who helped as we worked through the problems.
On top of that, thank you to everyone for being patient while the API was down (it ended up being more than 3 straight days). This problem gave me a lot of time to think about how to avoid a similar problem in the future, so hopefully we won’t be going down this path again any time soon!