ON an automatic, the o/d is controlled by that button and by the transmission and kickdown cable. The car starting default position is o/d engaged. It won't actually engage until the engine load and road speed are appropriate.
When you push the button you inhibit the trans from engaging.
The relay circuit that controls the amber arrow is separate from that which controls the solenoid on the tranny. The arrow can perform OK, and the tranny not.
It is likely that your o/d relay is failing, given that the arrow stays on when it should not. Of course, the problem could lie in the wiring to the shifter button, but that's less likely.
If you are able, the resoldering of the o/d relay is pretty simple. There is a post, Randy Starkie I think, that details the procedure and has pictures.
The labor to remove and install a relay is the same for new or resoldered one, and is quite small. Both require removal of the glove box, unclipping of the relay and removal of the connector. Budgetary constraints often require resoldering rather than installing a new relay. Your call.
Good Luck,
Bob
:>)
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