After much frustration and cursing, I've made some progress. Thanks for the suggestions!
Multiple anti-viruses, including those run with USB boots, found nothing.
I was never able to get the computer to start in Safe Mode. (This was a Windows 8 machine originally, an all in one, and when we upgraded to Windows 10, it seemed to have triggered a flashing screen issue that is most prevalent before Windows 10 starts. In the BIOS setup, it doesn't flash, but in all the 'options' for starting, and repair screens, it flashes on and off so that you're trying to read options a couple of letters at a time). I could never find a '4' option in the restart, or a point where F4 did anything.
Once the scans found nothing, I sat down and started killing processes, manually, one at a time, and then tested by trying to go to the Avast website.
The one that was blocking sites was identifying itself as Killer Network Service. Stopping that process allowed me to reach the antivirus sites.
This was showing in my Windows Startup as a 'disabled' startup, but it was starting up anyway.
After checking out 'Killer Network' on another computer, and finding some info that some malware identifies as the program, I deleted the folder and the shortcuts identified and everything I could find that was related, and emptied my recycle bin.
After rebooting, at least so far, I'm not getting automatically blocked from the anti-virus sites.
I found Killer Network Suite in my Control Panel/Programs and Features. Trying to uninstall brought up an install screen with a prompt to disable antivirus software. (It looked funky as hell rather than professionally done).
Using RegEdit to help, I tracked down every instance of Killer Network I could find in my computer -- files, temp files, and registration keys and deleted them all.
Now that that's done, I'll keep working on trying to get into Safe mode and then run scans again.
Mostly just mentioning it again in case anyone else has a similar problem, and finds 'Killer Network' on their computer.
