Moved Deskjet 3512 to a new room, updated computer OS, got new modem can only print through USB

[Lrblack]

In the spring we moved our printer from one room to another.  At first everything worked the same as it did before.  Soon after, anytime I wanted to print something I would have to unplug the printer then plug it back up.  If I had multiple jobs, I would have to do this with EVERY request.  In November I began having issues with my modem.  I contacted my provider and was sent a new modem.  I have also had a couple of updates to my OS recently.  Now I am having issues with my printer.  My printer shows that it is connected to my WiFi.  My WiFi shows my printer connected. However, I can only print if connected through a USB cable.   When I print my HP Network Configuration Page, it shows that the printer is not connected to the internet but it shows my computer and Hardware address for that computer along with my network (WiFi) name along with Signal Quality and Authentication Type (WPA-PSK).  I have reset my factory defaults on my printer. I have reconnected the printer to my WiFi.  I have checked for updates to my printer.  I have used the HP uninstaller.  I have deleted my printer from my Printer and Scanners of my computer along with trying to reinstall.  I have downloaded HP Easy Start.  I have installed the HP Smart App and attempted to use it to install.  I have even tried to install it using the HP Smart app on my iPhone.   I have done a series of steps which include plugging my modem, unplugging the power cord from my printer, then unplugging the the plug from the wall, waiting 60  seconds then plugging everything back up.  Nothing has worked so far.  I have spent many days, all day, trying to restore my once perfectly working printer.  

Any help would be so appreciated.