We have some progress, but first let me explain what's going on, in the hopes that someone here has some technical expertise and can make helpful suggestions (via comment or PM).
Our hosting service provides the internet residence for TMP, as well as our email service (on a mailhost we'll call Cyber). 'Automatic' emails are sent by the server communicating with Cyber. Staffers can also send and receive emails through Cyber, using email software (I use Thunderbird) or a web access portal.
When our hosting service moved our server to a new data center (apparently as part of a merger), the server lost the ability to communicate with Cyber. We were getting 'POP before' error messages.
Meanwhile, our hosting service introduced a new mailserver, which we'll call Janus. When this became fully operational, Thunderbird lost the ability to access email. Using the new Janus settings, Thunderbird is able to receive email only, and it is accessing email on the old Cyber service.
Simultaneously, for unknown reasons, the server regained the ability to send email via Cyber. The server cannot connect to the Janus mailserver, getting HELO (no such server) errors. This is also when Google/gmail began bouncing all of our server emails.
I switched to trying the web portal to access email on Janus, and the server people finally granted me access to manage email accounts on Janus. I discovered that all my old email was in one account, and my new email was in a different account (both on Janus). Janus is not receiving anything from the server, not even the bouncebacks from Google/gmail.
On Cyber, TMP had several email accounts that were aliases – that is, they looked like separate email accounts, but just fed into another email account. Janus does not allow aliases. It is not clear right now where email sent to those alias accounts is going, but it should be possible to set up actual email accounts for the old aliases, and set them to forward to a main email account. But to send from a former alias account probably won't work.
Bottom line: Looks like I can send and receive email from the TMP accounts through the web portal, though they need some adjustments. The server can send email via Cyber, with the Google/gmail problems mentioned, and not sure what will happen when Cyber shuts down.
One more wrinkle: Some of our server scripts send email via the Linux mail command. These are situation reports and are sent to my TMP email. For some unknown reason, they go through a mailserver called Carina, using a TMP email account that exists nowhere else. This was working, but has now stopped working.