Here is the best advice I can give you - cover with some kind of clear cover - a baggie, a plastic cup or container, a glass bowl or vase, or a glass tank ($10 for a ten gallon at your local WM store). Don't quit watering but do let it dry out a bit before you water again.

As long as you have plump rhizomes, you will be okay.

Here are some examples of mine:

The second one from the bottom is 'Purple Snow'. I lost that one because I forgot to water it but I have two more to replace it. Here is one grown outdoors for summer a couple of years ago.

To grow a big one try it in an aquarium. You might be able to put two medium sized begonias in a ten gallon aquarium. Aquariums have their own set of problems such as too much water, algae, etc. I have several aquariums with ferns and moss which I let them be. My big aquarium has been overtaken with episcia (this thing is a weed if grown in the right environment).

Rajah is a great terrarium begonia. It does very well for me if you keep the humidity high and water when it needs it.

Here is a recent picture of one from Logee's.

I've got the same thing (New York Swirl). I lost my first one so the second time I ordered it I put it in a container soon after I got it.

Here it is on the right with Maid Marion on the left.

They have been in these containers nearly a year and I may water them every 3 months or thereabouts.

I hope I can post it this time. I got rejected just a moment ago because GW says I already posted (the vagaries of GW).