What is the right way to clear the drilled plug out of a hole saw?
Ideally, hole saws have holes or slots in their sides, so you can pry out the scrap, or in the top so you can push it out.

(source: mkmorse.com)
Poke in there with a screw driver or other sharp object.  You shouldn't need to disassemble everything; just unplug the drill.
I assume yours don't have this already.  If you are going to be doing this a lot, you might want to buy a new hole saw with better holes.
Alternatively, you can try drilling holes in the top of yours.  Secure the hole saw in a vice or otherwise clamp it down firmly, and drill two holes on opposite sides in the top.
With two holes in the top, you can just push out the scrap with two large pins or nails.
Edit: some have larger side holes which may be easier to clear out.  You could possibly enlarge the holes on yours with a file.