Hm, as long as you're not seeing mushy stems I wouldn't worry too much, Cheryl. Give the stems a squeeze down near the bases, and if they're still firm, the plant is ok. I have lots of cane begonias growing outside in pots, and haven't ever lost one to overwatering. Last summer when a tropical storm brushed by, we got 9in of rain in 2 days. No problems.
Maybe it needs a little fertilizer? All that rain at once may have washed all the nutrients away?
The watermarks are sort of curious, I've never seen that before. All my cane begonias have dark or variegated leaves though so it might not show up.