internal fuse on air handler blows every time it shuts off
Are you 100% certain that it blows when the thermostat opens the call for cooling circuit to shut it down?  Or, does it run for a good while, shut down, and then you notice that the fuse is blown?  If I were you, before going down that rabbit hole I would put one more fuse in it, run it for 5 minutes or less, shut it down from the tstat yourself, and see if the fuse is blown.

If you do find that the fuse blows as soon as the tstat shuts it down, one reason it would blow after the call for cooling is gone is perhaps there is a blower keepalive timer (common in newer HVAC systems to push the last of the cold air out of the ducts) and that is somehow shorted out, causing it to blow the fuse when that circuit kicks on to run the blower after the tstat tells it to stop.