On re-reading, I see what I wrote made little sense due to autocorrection changing words. In the future I will try to proofread my posts better. 😊
Where I wrote
the accounts hat have been asking for passwords in the last few weeks have all been from the mail server formerly known as Verizon
I meant to say "...in the last few weeks have all been moved from the mail server..."
And where I wrote
I assumed the issue had to do with Verizon buying all and switching many of its email customers, including me, to all servers.
I meant to say "I assumed the issue had to do with Verizon buying AOL and switching many of its email customers, including me, to AOL servers.
Further, where I wrote
... when I ignore the problem it seems to love itself pretty soon ...
I meant to say "..when I ignore the problem it seems to solve itself pretty soon..."
Overall, I meant to say I thought the problems I saw were due to issues in the new AOL server, not my system. I did nothing and after a while, like maybe a month or two, the problem went away, mostly.
Still occasionally, but much much less frequently, an account asks for a password. If I provide the password in the dialog which asks, it is not accepted. If I enter the password in Internet Accounts it is accepted, usually. If it is for an account which I seldom use, I ignore the request and with no change from me eventually the server accepts the password.
To me, this sounds like the problem was mostly, perhaps all, with the AOL servers. In my case, the problem seems fixed, mostly.
I am posting this revised and extended comment in answer to more recent posts which ask "did that fix it?"