Can't verify identity server for gmail

you can update to MacOS 13 Monterey --But you will probably regret that.

With only 8GB RAM, MacOS after the Big Sur 11 version you are running now, have trouble running more than one simple App with appropriate responsiveness.

you DON'T need gmail to share files. if you can connect both Macs to the Internet at the same time without re-cabling, you have a highway already. it goes from each computer to your Router. Your Router can also act as an internet Switch, and send data back and forth between computers.

on either Mac (or both) use the Sharing control panel to enable File Sharing, and if you like Screen Sharing as well.

file sharing lets you mount the drive (or a subset of the files) on the other computer, accessible with the Network icon and providing login credentials for the REMOTE computer.

Screen sharing lets you have the other computer's screen appear in a window on the local computer, and you can take over typing and mousing as well.

Share the screen of another Mac - Apple Support (MK)

Set up file sharing on Mac - Apple Support (MK)

Share your Mac screen, files, and services with other users on your network - Apple Support (MK)

what happens is that Mail only recognizes ONE error: failure of the server to respond. if that happens while they are making backups, and you use:

Mail > mailbox > take all accounts ON line later, it may reconnect and work.

IF and ONLY if you know the password, you can use WebMail -- a very fancy term for a very simple operation: you surf with a web browser to the site that holds this messages and log in. Then ask t to show your emails.

once that is working, getting Mail working again requires a leap of faith:

IMAP protocol holds all email bodies on the server. When you connect with Mail, it shows you ONLY the headers in a list. When you select one to read, it sends a copy of the entire email to you. The original message is still on the server.

So IF and only IF you have the password, and have already logged using WebMail to be sure its right, and a reasonably modern computer, like MacOS 10.11 or newer, you suffer no harm in deleting the mail account off your Mail and adding it back in.

This process will open a webpage, and bring in other ways of verifications, such as email or text message to another account. Once fully re-verified, you will have access to these emails in Mail again.

<<. Thanks Grant! Now, if you could please tell me how/where I go to update to ONLY Monterey? When I click on Update, it seems to want to just jump to OS 18??? >>

I really suggest you NOT upgrade, because your computer will slow to a crawl and page incessantly. Apple no longer sells MacBook Pro or Air with only 8GB RAM memory because its is simply not enough for later MacOS.

YOU MUST have Trusted Backup to do any upgrades, because one misstep and your computer can only go BACK by Erasing and re-installing.