Begonia. What now?

I've cut off the stems. Some of the remaining stumps are well over an inch thick
DO I let the stems dry out, cut off some more or what?

 

I usually cut back, perhaps not quite as much as you have, I hope you stopped watering a while ago?
I then just put somewhere frost free under the greenhouse bench or in the garage and wait for the old stems to fall off, you can try a gentle tug, but I like to let them come away pretty much on their own.
Just tidy it up and keep it frost free, dry and cool.

 

Thanks @pete . I'll do that
God knows what size it'll be if it survives to next year!

 

I take mine out of the compost and scrape most of it off. Eventually the stems drop off. Then I put them all together loose in a large plastic pot either in the greenhouse or garage if it's below zero in the greenhouse.

 

I used to do that but I'm too lazy these days, must admit if there are any vine weevils in there you will find them that way.