The "Daily Limit for Unauthenticated Use Exceeded. Continued use requires signup." is misleading. It implies you've reached some kind of quota, but that is almost never the case.

The Drive quota for unauthenticated use is ZERO. So what the message is really telling you is that your Drive request isn't carrying an Auth token. You should trace the http and you will find that the requests that are failing are missing the Authorization header. As to why, well that's an exercise for the reader.