A faulty disconnector won't lead to full auto fire. It'll lead to hammer follow. Hammer follow will NOT lead to firing another round.
If I recall correctly, the sear is the component you need to worry about here. (You can look up how the 1911 design works online.) If it's functioning properly, the half cock will engage the sear and halt the hammer even if physical shock is causing the hammer to drop.
This problem COULD be intermittent in nature...meaning you may or may not be able to reproduce it "upon demand". Which makes it doubly dangerous, because a .45 firing multiple rounds unexpectedly will lead to DANGEROUS muzzle rise.
I doubt it's a simple cleaning problem. However, you may certainly tear down the pistol and inspect/clean everything.
But you really should have the gun inspected and repaired by a qualified gunsmith. This is NOT a safe firearm at this point in time.