I updated two machines to Unity 2017.3 and am getting Trello API: 503 errors on both of them. Still working on Unity 2017.1
Sorry for the delay in getting to this! I've been travelling with very limited internet.
It looks like 2017.3 came with some changes to the WebRequest API, which may be causing those errors to be raised erroneously. If Unity prompted you to upgrade obsolete API references in your project when migrating from 2017.1 to 2017.3, it's likely that some of the web request stuff in EF was broken by that. I'll put together a patch for the new API, do some testing, and get a fixed build put up by next week at the latest.
Thanks again for letting me know about this issue!