I haven't grown tuberous begonias since I left a part of the country where I could, but as a guess, I wonder how moist your soil is--could be too wet, rather than too dry as you mention--and what kind of soil you are using. Sounds like it may be rotting the plants--too moist, or not draining well enough, or not sterilized so they 'damp-off'. Does that sound possible?

These are tricky. Try to keep them out-doors/windows in shade first. After a few years, you will know what they want - basically cool nights, warm days, lots of bright light but not burning noon sun. Proper moist and fertilizing a plus.