We have a van that when you turn the key to the on position, it will start itself up and the starter will stay running after it starts. Shut the key off and the starter stops with the engine. Any one ever see one do this?
This is a 1997 E350 clubwagon 5.4

 

Pull the wire with the rubber boot off of the starter relay & I bet it wont do this. Looks like a short in the ignition switch.

 

We did find the problem to be in the neutral saftey switch on the transmission.
When we started looking more closely at it we found, the main wire to the starter is always hot so this eliminated the starter as the problem. We then put the shifter in to drive and turned on the power starter did not engage this way, I then shifted it to neutral and the starter turned over. The ingnition checked out good and so the only thing left was the switch on the transmission. I had to have something go bad in it. Put a new one in and all is good now.
I have just never seen one do it this way before. Mos of the time the starter quit working or they keep turning.