The Green LEDs you mention are simply connected to one of the motor windings.
They flicker, mostly for entertainment. If the motor stops rotating with a particular winding in a reverse polarity state, the LED will be extinguished, but motor still energized. Summary - flicker is normal.
I see you are already trying $4mi=1, I was going to suggest that.
What command are you issuing to get the motor to move?
e.g. G0 A3000, expecting about 10 stepper revolutions?
If I assume that A rotated normally when disconnected , it does sound like the stepper does not have enough torque to overcome the load presented by the mechanism.
First thing, I watched this https://www.youtube.com/watch?v=K_r8y_T8rsY which got me to thinking maybe I'd been setting my $afr and $avm waaaay too low. So I tried that and it moved stronger but still the same nonsense. Then I realized that I'd never actually verified that I had matching pairs of windings for A. I'd verified X, and Y&Z worked with the same color scheme so I assumed A was correct as well. I remembered that I had some issues out of A with Mach3 and LinuxCNC but I had other issues out of both of those so I just figured it was also with them. I think someone wired my stepper connector wrong because there's no way that the driver for A is wired that differently.
Sorry to drag you through something that I should know is a n00b error. They swore they tested this thing at the factory, I'm not convinced of that.