Let me tell you what is happening. I'm not sure if this can be adjusted or not.
I have an 87 IROC with a stock 700R4 and the stock shifter.
This is how I think it should work which in reality might not be correct.
To down shift from 4 to 3 or 3 to 2 or 2 to 1 you should have to push the button in order to move the shift lever correct?
Here is my problem. When I have the shifter in 2nd and I pull back on the shifter the car will shift down into first even if I don't push the button. I can also feel a click in the shifter. However if I let go of the shifter it will go back into 2nd. If I push the button it will go into 1st and stay there like I want. Sometime I will drive around in town in second gear when going from stop sign to stop sign. I drive with my hand on the shifter and sometimes it gets pulled into first gear even though I didn't push the button. I obviously don't want that to happen. How can I keep the shifter from going into first until I push the button. Is there any way to adjust this. Hope I am making sense.

 

Try adjusting the cable where it bolts on to the manual shaft on the transmission.