Check the wire connector on the side of the OD.  If the connector is tight I
would check the wire for voltage with a multimeter in various switch
positions.  You could run a wire under the door to check operation when in
motion or test on a hoist(CAREFULLY).  If the wire is functioning correctly
you will need to test the solonoid but I do not know how to do this.  Good
time to find a real GM manual, ask at a Vette dealer.

If you mean that the 4sp trans is popping out of gear, this is not a good
sign.  I would suspect (aggressive shifting caused) worn thrust washers in
the tranny or damaged shift rings but would want to check the adjustment of
the shift linkage first.

I would also want to change the oil in the trans (save the oil in a clean
container and search for metal both with a magnet and visually after
straining with a coffee filter.  Do the same change for the Dexron fluid in
the OD.  If the trans seems overfilled suspect the input driveshaft seal of
the OD leaking into the main trans.  This happened to mine at 70,000 miles.

My current (1987 Z51) OD problem is with too easy kickdown.  The OD kicks
down under part throttle when I want it to stay in high gear.  I am
considering rewiring the OD switch to operate full manual with no auto
kickdown.  I also suspect it may be an adjustment issue with the throttle
position sensor.  If anyone has done the switch rewire to full manual or
throttle sensor adjustment I would appreciate guidance.

It has been suggested that the best fix is a new Richmond 6 speed trans as
the OD is expensive to repair.  In my case the reseal job was not expensive
and it was good that I removed the tranny because the clutch was down to the
rivets but was not slipping.

Happy trails to all.

I would think the tranny pop could be solved with a rod adjustment.

As far as the OD, change the oil AND FILTER right now, and do it every
year.  Couple be a couple of things with the OD.  Here's how it works:

When you press the button, it's a momentary switch that sends a signal
to the ECM telling it to toggle the current state of the OD.  The ECM
determines if it should be on or off, and sends that signal to the
overdrive relay.  When the relay energizes, it energizes, the light
comes on the dash, and it energizes a solenoid in the valve body of the
transmission.    A valve body is just hydraulic "switches" that open
and close to route the transmission oil to the proper path to put it in
the gear required.

Since the light is coming on, we know the switch, ECM, and relay is
working.  The question is:  Is the signal getting to the solenoid.  If
yes, is the solenoid switching?  If no, then there could be debris in
the solenoid that is preventing it from moving.  The solenoid is just a
piston in a cylinder (it's only about 1/2" in diameter, if I remember
right), that is spring loaded.