You are sort of lucky in the way the failure that you are experiencing as normally it is the engine cooling control circuit that causes problems - and that leaves you with a fan running continuously even after the ignition has been switched off.
So, from what you have written down so far, I'd think that it is the AC side of things that is causing this
problem, fix that and the fan should stop running all the time.
Maybe, try unplugging the AC gas pressure sensor, you will find that on a metal pipe on the RHS of the car (LHS when looking back into the under bonnet area) just back from the coolant reservoir tank - I think, you will need to push that connector in slightly and then press down on its locking latch, then pull it off its mating socket - just a outside chance that that might do something, though I'd think that the
problem is further back in the AC controller.