Just dry them off, label them and store in a frost free place. I use onion nets and hang them up. You can use a shoe box, but no medium needed,as they could go mouldy.

When I used to grow these, I used to keep them dry and frost free, just in an open box on a shelf. No medium at all. They need checking for vine weevil grubs from time to time, and don't leave them where mice could get at them. I guess you could keep them in netting (from packs of oranges, for example) like dahlia tubers or canna rhizomes.

Snap, Dorjac!

I have a big orange begonia surviving a 3 or 4 seasons. I nearly threw it out as it was small. It grew huge in a basket on top of a chimney pot. I saw it had taken a dive out of the basket and lying on the lawn. I scooped it up. It had no roots worth speaking of, or much of a corm. I Trowelled a decent hole in the border and planted it quite deep for support. Still growing merrily. How on earth do these plants grow so much top growth and flower from such a small corm, in so few weeks? A really worthwhile plant to save.