Car battery stays on even when car is completely shut off?

Had the same problem a year ago, I would completely shut off the car but the ac light would stay on and you were able to turn on the ac. I would have to disconnect the battery every time I shut off the car so it would not discharge. This problem was also accompanied by poor idling at the time, both problems went away on their own but have now come back. This morning I turned on the car and immediately the rpm started going down and the car started shaking, every time I came to a completely stop the rpm went down and the car stalled. The car accelerates fine and stays on while accelerating but I could not reach my destination without having to place the car in Neutral when coming towards a stop and revving the engine to prevent it from stalling. When I finally arrived to my location and shut off the car to my surprise the ac light stayed on and the ac was able to be turned on even with the car being completely shut off. What could be the source of this or these problems and why do they always appear together? I checked the battery, alternator, and starter which all turned out good. I was told I could have an electrical short somewhere but if this is the problem where do I find where the short is? Any info on this problem would be very appreciated.

(1996 Honda Accord EX, Automatic transmission, F22B1 Engine)

Best to have a mechanic look at the problems to troubleshoot

Possibly a relay that supplies power to the AC sytem is sticking closed.

Try and find a diagrem of the fuse / relay board and see which one controls the AirCon and replace that.

Sounds like a computer problem.

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