Outlook 2010 Need Password button lights up and is unresponsive

I have a weird situation that has recently popped up with Outlook 2010 connecting to an Exchange 2016 Server that is offsite and in a different AD domain.  The Exchange Server is hosted by a parent company and all of my users have separate accounts in that domain for Exchange use only.  I know that isn't ideal but I have to work with it.  Anyway, this problem seems to have popped up after we enabled caching mode on the Outlook 2010 clients.  They had previously been directly connected for years but people have been complaining about slowness so our answer was to use cached mode.  

What is happening is this:
The user will start Outlook 2010 and be shown the cached contents of their mailbox.  The Exchange server will try connecting and then show a NEED PASSWORD button where it would normally show Connected to: Microsoft Exchange.  You can click the button all day long and nothing will pop up letting you put in credentials.  If I delete the Outlook profile and recreate it using the Outlook 2010 wizard, it won't even resolve the name to a mailbox after it knows what it is looking for in autodiscover.  I can have them log into OWA just fine but Outlook 2010 won't find the server or mailbox at all.  The only way I can recover from this is to delete the user's Windows 7 profile and start them with a fresh one.  Only then will it resolve the name on the server and connect the mailbox after supplying the proper credentials in that domain.  It may work for a day or two or even weeks until it displays NEED PASSWORD again at which time it becomes broken until you reset the entire Windows profile.  As a side note, the password has not been changed in the domain in which the Exchange Server exists when the problem presents.  It just pops up for no apparent reason and it is only since enabling cached mode that we have noticed it.  I find it so weird that once this happens it won't even resolve the mailbox until you completely wipe the Windows profile and start with a fresh one.  This is not ideal.

These two things seem unrelated to me ... using cached mode and not being able to connect but I can't write it off as caching not being the root of the problem in some way.  I am just curious whether there is a bug of some sort that is triggered by this perfect storm (Outlook 2010  in cached mode, Exchange Server 2016 in a different domain)
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Steve BantzIT ManagerAsked:
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efrimpolCommented:
Just curious. Are you authenticating to the offsite Exchange Server using the end-user's profile name or the email address?

Our Exchange Server is also hosted and we require that the username is actually the email address.
Steve BantzIT ManagerAuthor Commented:
We are using domainname\username format.  Is email address format the proper way?  It had worked using the former format UNTIL we put the clients into cached mode.  I will try the email address and see if that changes anything.
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