I've discovered that I've now got hundreds of tiny hair like micro-spines in
my hands.

Those are called glochids. They have hooks on the end. They are unique to the
genus Opuntia and its relatives. The more advanced cacti have only straight
honest spines.
You probably should see a dermatologist.
Iris,
Central NY, Zone 5a, Sunset Zone 40
"A tree never hits an automobile except in self defense." - Woody Allen