I, myself, am fairly new with begonias. I have a new Angel-wing that I'm falling in love with. But, I'm thinking that it could be the combination of things that you spoke of. Furnace/Dry heat, lack of humidity…So I'm sure that wonderful pebble tray may help…A regular misting, and only watering as needed instead of on a regular weekly schedule may help as well. I think, although I could be wrong,
but I believe all plants tend to go into a little bit of shock, especially when they must be moved suddenly to a different 'climate'. So I'm thinking the pebble tray, and trying to emulate the light that it was getting may help. I hope!