Can I Recover Lost Emails on iPad After Changing Password?
After being told on the Apple Discussion boards that this can't be done, I found a solution.

I re-entered all the original log in information into the account details.  It took a bit to get them to stick, because iPad kept trying to verify the account, which it couldn't do, because the legacy server wasn't responding, but I finally got it.  Once the I had the original credentials entered, all the original Inbox and Sent emails were back.

One important key: Although I didn't change my email address, the new server did require a different username syntax.  The old username was just "name" while the new username was "name@mydomain.com".  That little change is what caused iPad to decide it was an entirely different account.