Thanks for the reply Eddy.  I followed the links and and did another complete uninstall and reinstall, (including additional cleaners EZ Cleaner, IOBIT uninstaller, and CCleaner, along with Avastclear).  Rebooted, reloaded, and opened Avast.  After running the Quickscan, I tried the Scan for Outdated Software, and the old "aswOfferTool.exe" warning box popped up.  I clicked on the close button and the box popped up twice more, requiring me to hit the close button each time, until the widow showing the scanned programs opened showing that they were all current.  This was the same problem I had previously.  

 I am in the process of researching new notebooks/tables/hybrids in order to replace the desktop I am currently using,  (old, slow HP circa 2006 running WinXP3).  I am going to just keep using this one as is and wait for my new unit to load a fresh install of Avast.  I'm probably going to turn this hard drive into an external storage drive, and then, problem solved.  Thank again for the effort.