Microsoft Support got this resolved! Our main issue was that we had a different version of AD FS than the instructions were written for. In addition to the AD FS Powershell command the system gave us we needed to use the following command:
Grant-AdfsApplicationPermission -AllowAllRegisteredClients -ServerRoleIdentifier "https://auth.'OurOrganization'.com/"
We also needed to tweak our IIS settings, which were as follows when it worked:
Server :
Anonymous Authentication Enabled
ASP.NET Impersonation Disabled
Forms Authentication Disabled
Windows Authentication Disabled
Microsoft Dynamics CRM (Site)
Anonymous Authentication Enabled
ASP.NET Impersonation Disabled
Forms Authentication Disabled
Windows Authentication Enabled
XRMServices
Anonymous Authentication Enabled
ASP.NET Impersonation Enabled
Forms Authentication Disabled
Windows Authentication Disabled
XRMServices > 2011
Anonymous Authentication Enabled
ASP.NET Impersonation Enabled
Forms Authentication Disabled
Windows Authentication Disabled
XRMServices > 2011 > Organization.svc
Anonymous Authentication Enabled
ASP.NET Impersonation Enabled
Forms Authentication Disabled
Windows Authentication Disabled
nga
Anonymous Authentication Enabled
ASP.NET Impersonation Enabled
Forms Authentication Disabled
Windows Authentication Disabled