Most begonias won't wait for a frost to begin dying back. They will begin fading once the night temps begin falling below 40. Unless to get a quick snap freeze, just leave them until they begin showing the distress, then cut them off flush. Pull the tubers and shake as much dirt off as you can. If possible, leave them sit on top of the soil for a few days before bagging them. Do NOT place them in sealed bags, punch some holes in the bag for air circulation and sawdust works fine. Just be sure it isn't bone dry. Don't want it damp, just misted a bit.

Just noticed here is a post similar to mine!

What kind of bags? Paper or plastic ? (Hmmm...where have I heard that before??)

I have no sawdust; how about perlite or vermiculite?