If it is an older printer, the waste ink pads, which sit at the base of the printer, are probably full of ink, and it is overflowing.
I HAVE FOUND EXACTLY THE SAME PROBLEM JUST RECENTLY ON A 6 YEAR OLD HP 812C. IT WAS STLL PRINTING EVERYTHING WITH NO PROBLEM BUT THERE IS A WET STICKY MESS OF BLACK INK IN THE WELL BELOW THE CARTRIDGE PARKING PLACE AND THAT HAD STARTED TO LEAK OUT OF THE BOTTOM OF THE PRINTER.
CAN ONE CLEAN IT UP AND REPLACE THE WASTE INK PADS OR IS THE ONLY ANSWER TO JUNK IT. COULDN'T FIND ANYTHING ON THE HP WEBSITE BUT THAT IS PROBABLY BECAUSE THEY ARE NO LONGER SELLING THAT MODEL.