Bill,
Fixit has a good suggestion. Check your signal lead from the new switch that you installed and see if it is hot all the time. If the solenoid is open with everything disconnected and then closes when you hook up the starter there must be a short in it or something in the new switch is operating it. Take both leads from the new switch off the solenoid and then hook your starter lead to it and see if it shows closed. There should be no reading at all between the post that the battery cable goes to and the one the starter goes to unless the solenoid is operated or shorted.
Take all leads off. Check between battery post and starter post on solenoid and your meter should show nothing. (open) Then hook up one of the small control leads and see if it closes. If not then hook up the other control lead and see if that operates the solenoid giving you continunity between battery & starter posts. If you get a reading now, then the solenoid is being operated by your control leads. If no reading then hook up starter lead and see if you get a reading through to the battery post. If you do, take one control lead at a time off and see if you lose the reading. I really kind of suspect you may have a hot signal wire as Fixit said or a faulty switch or solenoid. The fact that you read it open when you took the wires off makes it sound as if the contacts aren't welded together but they could be burned enough by the things happening to cause trouble later on. Good luck and keep us posted.
Ron
