Solution 1:

I was able to fix this myself. It turns out that the local machine accounts had cached bad credentials that were (correctly) failing. Thanks to @greg-askew for pointing me in the right direction. For anyone who stumbles on this looking for a solution:

  1. Download PsExec (part of sysinternals) from https://technet.microsoft.com/en-us/sysinternals/pxexec.aspx
  2. From an elevated command prompt run PsExec.exe -i -s cmd.exe (This will open another command window in the Local Machine account context).
  3. From that window, run rundll32.exe keymgr.dll, KRShowKeyMgr (That will open a gui with a list of cached credentials).
  4. In that gui, delete any credentials that look suspicious (In my case the credentials were named after my PDC).

After deleting the credentials from the cache, it immediately started working again.