Turns out the problem was the 'error reports' (as opposed to log files) that were being generated filled up all available disc space and were making the server unstable and eventually crash. One of the crashes must have happened in the middle of the AGS writing an important .xml file. I Tried troubleshooting with ESRI tech support, but ended up reinstalling the server software.
I sent ESRI one of the 'error reports' - hopefully they can tell me why multiple 5+GB files were being written. In the meantime, I changed the settings to not save any error reports.
