Cars of that vintage did indeed lock at speed (15-20 mph in gear) and did not automatically unlock when the engine is shut off. My wife's last car, a 97 Camry, and her current car, 2012 Chrysler 200, both unlock when shut off, but they both had alarm systems that I think altered their behavior.
As to locking them with the unlock button, either the switch is faulty, or the Body Control Computer may be caught in a latchup condition. I had this happen when I got my car back from painting, and they had had the door panels and switches out. When I got it back, the lock button was not fully snapped into the door panel, and it misbehaved like this. I unplugged and plugged the switch a few times, disconnected the battery and the BCM, and finally it cured itself. Not sure which specific action did it, but I never had trouble again, 10 years later when I got rid of it.