The vehicle uses Shift On The Fly engagement.
The transfer case gets engaged, spinning the front driveshaft. Then two halves of the front diff get connected by the actuator.
Coming to a stop, stopping, "put(ting) the truck in neutral, push 4 hi, put it in gear and roll forward a bit but the 4x4 doesn’t engage"? It doesn't work that way.
My '88 has the lever on the floor, same with my '89, push button on my '00. Any of those, I drive any speed above 15 mph, let off the gas pedal, push the button, press the gas pedal again. Click. In 4x4.
Now, the flashing indicator tells me that there's another problem. And you're gonna need some diag time with a wiring schematic and a test light. Start with the t-case position sensor, then test for power at the TLA or worm gear actuator ('97 was a transition year, not sure which you have...)