Well, the problem seems to have resolved itself, possibly after another update from MS.
I don't have Pegasus installed in anything like what was just suggested. I have separate partitions set up on my PC for the OS, my applications, and our home directories. Pegasus is installed with my other applications, in the D: drive. My Pegasus mailbox is installed in my home directory: E:\<My Name>\PMail. This has worked perfectly for me for over 20 years, but Windows 10 is clearly making it for difficult for the applications.
The problem yesterday seems to have been that non-MS applications, that wanted to check on their licenses when starting, were having trouble getting an internet connection. This affected HR Block and Quicken. It also seems to have affected GlobalProtect, which needs to make a VPN connection to the university where I worked to give the OK on the license for the Windows 10 Enterprise that I'm using. Normally that needs a refresh about every 6 months, so it's not something that I get really familiar with. Yesterday GP would not connect. This has never happened before.
Everything seems to be working this morning, but I noticed one change: When I brought up GlobalProtect today, I got a login page that I didn't get yesterday. Yesterday, when I brought up GP, I didn't get the login page, so GP didn't get my password. When I ran the troubleshooting logging for GP, I saw it complain about no password, but I wasn't sure if that was normal or not, having never watched the logging before. Today, I got a completely different GP first screen which came up with my username and asked for my password. After that, it connected and now everything seems to work correctly.
Pegasus now connects to my POP3 and SMTP servers within 1-2 seconds, just as it always has, so things are back to normal. Frankly, I don't know what fixed it: the MS update, the change in GP behavior, or the simple fact that there was another reboot in there.