From your post I assume your car has not a tachometer, so you cannot see the engine speed. right?

The green arrow lit means that the auto transmission is inhibited from shifting into overdrive. It matters not what gear the trans is in at the time. Light off is the default position upon starting the engine.

Light off means that when the engine speed, road speed and engine load are correct, the trans will engage the overdrive. Light on means it won't engage.
I found that on a flat road at about 40 mph the overdrive engaging will drop the engine rpm by about 20%, with no change in road speed. Good for fuel saving.

With OD engaged, pressing the gas pedal down a small bit will drop OD out, thus raising engine rpm to a level where more power is available. Nice for passing.

Your problem is likely a wiring situation.

To learn if it is the OD relay, if all things (speed,load,road) indicate that OD should be engaged and it is not, with the heel of your right hand whack the dash just to the left of the glove box door. The relay is just behind that spot, and can be a little sticky with age and a little jarring can un-stick it. BTDT

Good Luck,

Bob

:>)