An empty black cartridge that thinks it has ink, may be a symptom of a stuck black-change valve.
There is a valve that switches the head between matte black and photo black. One of the he failure modes allows black ink to leak out, emptying the cart. The chip doesn't know about the leak, and thinks the cart still has ink left.
If your replacement black ink cart quickly exhibits the same symptom, then I would strongly suspect that this is your problem.
Sometimes switching black inks can unstick the valve.
Remember, even if you never switch black inks, the 3880 will occasionally initiate a switch for a cleaning.
The 3880 failures I see most often are failure of this valve, and failure of the networking port.