Begonias only producing single flowers
Thanks for your question about begonias. It is certainly possible that too much shade would reduce flowering, but it sounds to me like you may be getting mostly female flowers (which are single) instead of the fancier males. You can tell by looking at the back of the flower for a three-sectioned ovary. Usually there is one male flower, flanked by two female flowers, which many people remove since they are less showy.
Unfortunately I have no clue why your plant might produce only female flowers.
Another possibility is that it is not the cultivar you thought you were getting, and has generally less showy flowers. You can examine the flowers - if both male and female are being produced, this is the most likely explanation.