1999 Ford Expedition Starter runs continuously after engine

One day the 1999 Expedition (4.6L) won’t start.
Turn key, lights dim (normal), something clicks (normal), nothing turns over.

After a lot of looking around and scratching our heads, a friend jumps me, it starts.

The non-start problem happens very frequently (2 out of 3 times) but I figure out I can short the starter relay to start the engine.

I replace the starter relay on the passenger side fire wall with one a friend had. Now the engine starts every time, but the starter won’t stop.

I try putting the old relay back on. Starter continues to run after engine starts.

I replace the starter with a new one. The new starter keeps running after the engine starts.

I buy a new relay. Starter still continues to run after engine starts.

I try to turn the key on (not start) position, and short the terminals on the relay. Engine starts, but starter continuously runs.

With key out, voltage across battery is 12vdc, voltage from starter to ground is 12vdc on one post and 0vdc on other.

Seems like something is telling the starter to keep going even though the engine has started. There are NO OBD2 codes present.

Suggestions?