My HP ink jet printer was making such a mess that I was ready to replace it. New cartridges didn't help, nor did my efforts at cleaning. A search of the HP support pages turned up explicit instructions about cleaning. After following those instructions, the printer is working like new.
If the ink is actually running down the page and not just being smeared across it, replacing the cartridge sounds like a good idea. Whether that helps or not, you should search the manufacturer's support pages for maintenance/cleaning instructions. You may also find pertinent troubleshooting help.
In my own situation, the key bits of information from HP were as follows: that distilled water should be used for cleaning, explicit instructions about what & where to clean, and that cleaning should be repeated as required (2 times for me). I also learned that it is not a good idea to leave the printer on nearly all the time. On the HP the cartridge parks on a rubber seal that keeps the ink fresh - and ink cartridges do age/expire even if unopenned. Finally, there is an ink reservoir for ink overflow. It contains a sponge that can become saturated and cause problems (maybe yours?). This is a very nasty mess.