I carried out the above strategy and all works now, after hours of screwing around and trying to find out how processes that have worked for years have suddenly become so weird this week. I thought I would report what I did here in case it might be helpful to anyone else who has these problems.
I uninstalled the previous HP Utility and HP Easy Scan apps, and removed their files from the Library file as well. Then went to the HP site, added the HP Smart app, Easy Start app, and redid computer firmware to boot. Then things started to work, but with another weird thing. When I would try to print or scan, the computer would identify files that would "damage" the computer and refuse to proceed. I deleted those files one by one. Now, it seems all is back to normal with printing and scanning.
This all came about for me just after macOS10.15.7 update this week. Apparently it isn't compatible with older (a few years) HP apps. While I agree this isn't Apple's fault, I do think it would have been nice if HP had alerted all of us and simply told us to reload newer versions of their apps. Would have saved me a lot of time. Too much to expect, I guess.
Interestingly, my iOS devices (iPad) still work fine with no interruption. I'm liking iOS better than macOS all the time.