Am I following the procedure to set the sensitivity in Try Mode for the A1362 Hall-Effect sensor correctly?
You're probably suffering from bounce.  If you were to connect an oscilloscope to the Vout pin (I suggest you try this) you might well see something like this:



When you plug or unplug the jumper, it will be almost impossible to do this without causing several unintended pulses. I recommend not trying to program it by using jumper wires.

By far the best way to do it is to use a microcontroller and some transistors. However, if you really want to do it by hand, using this circuit might help:



Here, you can press S1 to pulse Vout to 27v, and S2 to pulse it to 3v. C1 will help to remove any switch bounce. I don't guarantee that this will work though.