Any help appreciated!

It does appear you have some stems that are rotting off which is a sign the plant has too much water. Cut back on watering - let the soil dry some before watering again. Outdoors this should not be an issue as the heat dries them out even if it has been raining quite a bit.

The leaves that show rot near the base of the rhizome can be cut off and restarted in many different ways so you would have a little insurance in case the mother plant croaks. Do it sooner than later.

I just moved a bunch of my starter plants from leaf wedges to the great outdoors a couple of weeks ago, hoping the squirrels let them alone.

I find Jiffy pellets the easiest way to do this.

after several weeks I then move them up to bigger pots (3 to 4 inch to begin with)

One failing leaf is not a cause for concern. The leaves of rhizomatous Begonias come and go. Your plant is about to bloom, and is growing new leaves. It can't be too terribly unhappy.

Looks like they were grown in death traps. Peel off the mesh surrounding them in the soil - they're intended to dissolve but they rarely do. And know that a begonia almost always readjusts once brought from nursery to indoors. It will probably grow back smaller leaves to accomodate the new, less light

Is your Begonia recovering?