the camera is staying on even though the alarm is off?

You'll need to be a bit more

You'll need to be a bit more clear with what you're asking.

The camera is always "ON"
Whether it's doing something is an entirely different thing.

If the shutter is open, then

If the shutter is open, then it is recording constantly on an internal loop. This provides the "pre-roll" function, which when the camera is triggered, gives video from a few seconds BEFORE the trigger. If the shutter is closed, the camera cannot record or transmit any video.

The camera only TRANSMITS video (for viewing or recording) when the alarm goes off, when another system event (test, arm, disarm, secret alert) occurs, when "live viewing" is requested, or when the camera motion detector is active (you select when this is) and senses motion.