The current in the starter circuit starts at the battery, passes through fusible links and fuses, then through the ignition switch (switch is closed when the key is at the START position) and then takes a tour over to the glovebox before coming into the NSS as the black-with-white-tracer wire. From there, assuming the transmission is in neutral or park, the current leaves the NSS as the black-with-red-tracer wire and goes to the starter solenoid. It is a fairly easy circuit to troubleshoot (when the problem is present).
In my case (as well as Romer's and TOY350's) the connectors on the two wires were faulty. The black/red trace wire was broken inside my NSS, and the black/white trace wasn't far behind. Your symptoms sound similar enough to mine/Romer/TOY350's that I would strongly suspect the NSS connector. The only other relatively complex part in the starter circuit is the ignition switch (not the tumblers) and it is clean and dry up inside the dash.
Spend half an hour (at most), unplug the connector off the transmission and examine the two larger gauge wires I mentioned above. Disconnect the two wires from the plastic connector body and examine them up close to make sure they aren't separated. This will either be your problem (my guess is that it is), or it will rule it out and let you check the next part of the circuit.