Hi! My name is ***** ***** I'll be assisting you today.
It sounds like the problem is with the ink carriage/printhead rather than the ink cartridges. Have you tried cleaning the printhead itself with qtips and rubbing alcohol?
Turn the printer off, take out the ink cartridges and clean the contacts where your ink cartridge and carriage make connection.
Another thing you can do is try rolling a small piece of paper into roll, then flatten it and hold it just inside the edge of the cartridge holder as you slide the cartridge into place, so that the thickness of the paper pushes the cartridge towards those contacts at the rear.
Let me know if this works or not. Just reply back to this and I will be able to answer you as soon as I can.
If it DOES work, please kindly rate my answer with 5-stars so that I may receive credit for assisting you today!