It is not just with the wifi because the hardwired connection from the router to the computer was also ineffective until the router was reset. So, somehow the router itself was losing its connection to the internet.
But I *think* I may have found the solution. For the last year we have been having problems with the wifi printer showing "offline" to the devices using it, and we would have to remove it and add it back. Then when we got the test modem/router, the printer couldn't be seen at all, and everything worked fine. So, I did some searching here and elsewhere and it seems that such printers being assigned IP addresses this can somehow interfere with the router's IP address, causing conflicts and disconnecting one or both. So, we hooked back up the router and and turned off the printer's wifi altogether and that SEEMS to have solved the problem. We have been connected for 24 hours straight, which has not been the case in a quite a while. Maybe this might help others if it ends up solving the problem. I will report back.