another thing to check is the starter relay in the underhood fuse box. try switching a relay from another [known good working] position. this, at least, will show if the relay is possibly taking a dump.
there is a diagnostic tree in the FSM that shows how to test the wiring at the starter relay position.
i have had a couple of caRRoT vans have had the same issues as you are experiencing. one, the relay was bad, the other had a bad wire connection, causing the relay to not function.
have had a ton of starter contactors over the years act up and burn/pit/weld together. they are easy to change.