Definitely sounds like the Neutral Start Switch (NSS).
The switch itself is fairly easy to rebuild yourself. Just don't lose any of the
tiny bits. Plenty of info on how to do this online.
Putting it back and re-aligning the switch is another matter.
The Shop Manual says to align the vertical groove on the valve shaft with a
"standard line" that is molded on the switch cover. Sounds easy, but for me,
this has always been a blind shot, hit-or-miss ordeal. It also often resulted
in me still having to jiggle the shifter handle in order to engage the starter.
Instead, I've found that using a "beep" continuity tester on the "B" and "C"
terminals on the switch harness to be more accurate. Most digital VOM
meters have a continuity tester built into them.
There are 8 terminals on the connector. Looking dead-on at the connector
end, with the retaining clip held at 12 o'clock up, the "B" and "C" terminals
are the two center terminals, lower row.
1) Set the transmission in "Neutral".
2) Slide the switch fully onto the transmission and install, and torque,
the Adjusting Bolt. This assures that the switch is fully seated and
square against the transmission body.
3) Install the lock washer, and torque Attaching Nut on the valve shaft.
4) Return to the Adjusting Bolt, and loosen it by 1-2 turns.
5) Connect the continuity tester to the switch harness terminals ("B" and
"C") and rock the switch body a few degrees, back and forth, using your
hand until you hear a "beep". Keep rocking the switch body back & forth
a little until you think you feel that you've found the "sweet spot" or
center-most position for obtaining the "beep".
6) Hold the switch body securely and tighten the Adjusting Bolt.
7) Get in the driver's seat and confirm everything by putting the transmission
shifter in every gear.
You should hear "beeps" only while in Park, and Neutral - without having
to jiggle and shake the shift handle.