Perhaps you can move the leaves to a dryer growing medium. Begonias do well if they are thoroughly watered and then left until the soil is almost dry about half way to two thirds down in the pot. My rex begonia does very well if I drench the soil which drains quickly, then I do not water again until it is quite dry. The stems are able to store water kinda like a succulent. Let the leaves get dry but not too dry and just mist them as needed. You could try using a moisture meter to check for dryness.
So far my leaves which are cut into pieces and placed into peat moss are ok. I watered the pete moss placed the container into a slightly opened clear container and have just misted them occasionally.
