Yahoo/Att mail issues

Earlier today I saw an error message on my MacBook Pro asking me to log into Internet accounts. My mail wasn't working. I had to update my password. Mail is now working on this computer; however, I have a second, older MacBook, and I can't get mail to work on it. I've tried going into Internet Accounts and entering the new password, but it keeps getting rejected. I tried an ATT help site, but it required me to first enter my email (the same address I've had for years), and the response was that my address couldn't be found, so I couldn't get any further. When I tried a chat, an automated system was answering, and it didn't provide anything that was useful. The upshot is that every way I try to resolve this situation I'm blocked by rejection of some piece of information, whether it's the password, user ID (which I assume is my email address), or something else I've done wrong that I can't identify. Surely there's a way out, but what?