This is a newer model than some of your printers likely, and that may have something to do with it as well, they purposely try to deny the use of other cartridges if they aren't an exact fit. The "faked" cartridges won't work in most cases, if it is a cartridge that was an original epson one that was just refilled, or a new cart with an oem chip in it, that is a different story though.
I still believe this to be just bad cartridges, and eventually you'll likely find a brand/model that will work.
Just to be safe though,
What we can try are some cleaning and reset steps.
First remove all of the ink cartridges, then clean the metal electrical contacts on the back, as well as where they mate up inside the printer, using qtips and rubbing alcohol. Dry well, then reseat the cartridges.
Then reset the machine:
unplug power at the wall outlet
wait 5 minutes
press and hold the power/on button for 60+ seconds
keep holding while plugging back in, and for another 60 seconds after, then release
Let me know the results, and I'll advise further if needed.