Bill,
Try disconnecting the cable from the starter to the solenoid and if you have a meter check to see if you have battery standing on that terminal when the battery is connected. All the solenoid amounts to is a relay that operates to cut the battery through to the starter which has ground on one side of the winding. I would be checking that solenoid even though it is new. I am sure that you have already tried all of this but like trouble shooting a computer program, you have to break it down into areas. Another possibility is the new switch you installed. Check the wires from it to the solenoid to make sure they aren't keeping the solenoid activated. An easy way to check the solenoid is to remove the two wires from the switch that activate it and then measure the side going to the starter with that lead removed. If you get battery there then the solenoid is operated or shorted.
Ron
