Non-Apple mail issues since Security Update 2019-001 installed - unable to connect to other emails since update

Since I installed Security Update 2019-001 on my Mini yesterday, I can no longer connect to non-Apple e-mail accounts, specifically yahoo.com and gmail.com. I am prompted to enter my password for each of the affected accounts, but with no results. For example, I just entered the password for one of my gmail accounts, and after it spins for a bit, the only thing showing is a blank pop-up screen with a header that says "Enter the password for the account "My google"" -- no place to enter the password (again) and only a Cancel button at the bottom of the screen. For Yahoo with two factor authentication, when I'm prompted to enter my password on the Mini, it brings up the authentication screen where it sends a notification to my mobile phone. I respond appropriately on the cell phone and Yahoo provides a code which I then type into the screen on the Mini. Then I get the rotating snowflake or whatever you want to call it, and it just keeps spinning. The only option here is to cancel. I have verified that the passwords are still okay with each vendor and can sign on to webmail with the same passwords I'm trying to enter in Apple Mail. The only conclusion I can come to is that something changed by the Security Update that is preventing me from logging in to these other e-mails. Hardware is a Late 2012 Mini running Sierra 10.12.6. Yes, I know I need to upgrade; it's in the plan. But this was working before the Security Update.