How does my 4x4 push button locker work in my 1989

That depends a lot on how your truck is equipped. GM uses several different 4 wheel systems including AWD, Part-Time 4WD, Full-Time 4WD, and ON-Demand AWD. Then there is another version used on some trucks that is a combination of 4WD and On Demand AWD. - it gets a little confusing but each system as slightly different charachteristics.

Genericly speaking, if you have the push-buttons on your dash that say "4-HI", "4-LO", and "2HI" and/or Auto-4WD , the transfer case has an electronicly-controlled stepper motor which shifts the transfer case into the selected mode. An electric solenoid/actuator then engages/disengages the front differential for 2-Wheel or 4-Wheel modes. One push of the button and it shifts the transfer case and engages the front differential.

That's much easier than the old days when you had to stop the truck, get out (Sometimes in the middle of a big mud hole) and lock the front hubs then get back in and play with the lever in the floor to get the transfer case to shift into the desired mode.